Nilpotent Group-Counterexamples to Zilbers Conjecture

TL;DR

This paper constructs specific 3-nilpotent groups that serve as counterexamples to Zilbers Conjecture, using graded Lie algebras over finite fields, expanding the understanding of model-theoretic properties of such groups.

Contribution

It introduces uncountably categorical 3-nilpotent groups of exponent p > 3 that are counterexamples to Zilbers Conjecture, employing the additive Collapse method.

Findings

Constructed uncountably categorical 3-nilpotent groups of exponent p > 3

Demonstrated these groups are not one-based and do not interpret infinite fields

Extended previous work on 2-nilpotent groups to 3-nilpotent case

Abstract

We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new uncoutably categorical groups were contructed in [3]. Here we use the method of the additive Collapse developed in [5]. Essentially we work with 3-nilpotent graded Lie algebras over the field with p elements.

Equations

$$Z_{c+1-i}(F(X))={\Gamma }_i(F(X))={\oplus }_{i\le j\le c}{U}_j(F(X)),$$

$${\delta }_i(M)=\sum _{1\le j\le i}o-di{m}_j(M)-(\sum _{2\le j\le i}ideal-di{m}_j(M))$$

$${\delta }_2(⟨AC⟩)\le {\delta }_2(A)+{\delta }_2(C)-{\delta }_2(A\cap C).$$

$$X_2^0(B)X_2^A(B)X_2^C(B)X_2^1(B)X_2^2(B).$$

$$⟨AC⟩\simeq F(X_1(AC)X_2(AC))/I,$$

$$\mid X_2^C(B)\mid +\mid X_2^1(B)\mid +\mid X_2^2(B)\mid +\mid X_2^1(A)\mid +\mid X_2^2(A)\mid +$$

$$\mid X_2^A(B)\mid +\mid X_2^1(B)\mid +\mid X_2^2(B)\mid +\mid X_2^1(C)\mid +\mid X_2^2(C)\mid -$$

$$\left(\mid X_2^C(B)\mid +\mid X_2^A(B)\mid +\mid X_2^1(B)\mid +\mid X_2^2(B)\mid +\mid X_2^0(B)\mid \right)=$$

$$\mid X_2^1(B)\mid +\mid X_2^2(B)\mid +\mid X_2^1(A)+\mid X_2^2(A)\mid +\mid X_2^1(C)\mid +\mid X_2^2(C)\mid -\mid X_2^0(B)\mid =$$

$$\mid X_2(AC)\mid +\mid X_2^1(B)X_2^1(A)X_2^1(C)\mid -\mid X_2^0(B)\mid =$$

$$o-di{m}_2(AC)+\mid X_2^1(B)X_2^1(A)X_2^1(C)\mid -\mid X_2^0(B)\mid .Hence$$

$$\mid I\mid -\mid I\cap F(X_1(B)X_1(A))\mid -\mid I\cap F(X_1(B)X_1(C))\mid +\mid I\cap F(X_1(B))\mid .$$

$${\delta }_2(AC)\le {\delta }_2(A)+{\delta }_2(C)-{\delta }_2(A\cap C).$$

$${\delta }_i(⟨D^{\prime }C⟩)\le {\delta }_i(D^{\prime })+{\delta }_i(C)-{\delta }_i(D^{\prime }\cap C)$$

$$0\le {\delta }_i(⟨D^{\prime }C⟩)-{\delta }_i(C)\le {\delta }_i(D^{\prime })-{\delta }_i(E\cap C),$$

$${\delta }_i(⟨E^{\prime }C⟩)\le {\delta }_i(E^{\prime })+{\delta }_i(C)-{\delta }_i(E^{\prime }\cap C).$$

$$0\le {\delta }_i(E^{\prime }C)-{\delta }_i(C)\le {\delta }_i(E^{\prime })-{\delta }_i(E^{\prime }\cap C)\le {\delta }_i(E^{\prime })-{\delta }_i(A).$$

$$\forall xy\left(U_i(x)\wedge U_j(y)\wedge x\ne 0\wedge y\ne 0\to ([x,y]\ne 0\vee \underset{k\in K,k\ne 0}{\bigvee }x=ky)\right).$$

$$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\otimes}_{B}C$$

$$⟨ABC⟩=⟨AB⟩{\otimes }_{⟨B⟩}⟨BC⟩.$$

$$X^B{X}^C{X}^A\{[x,y]:x\in X_1^C,y\in X_1^A\}$$

$$X^B{X}^C{X}^A\{[x,y]:x\in X_1^C,y\in X_1^A\}\{[w,z]:w\in X_2^C,z\in X_1^{A}orw\in X_2^A,z\in X_1^C\}$$

$$\{[[x,y],z]:x=z\in X_1^C,y\in X_1^{A}or$$

$$x=z\in X_1^A,y\in X_1^{C}orx<z\in X_1^C,y\in X_1^{A}orx<z\in X_1^A,y\in X_1^C\}$$

$$e=\sum _j[a_1^j,v_1^j]+v_2,$$

$$e=\Delta +v_3,$$

$$Y_3^B=\{[x,y]:x\in Y_2^D{\bar{e}}_2{\overline{e^{\prime }}}_2,y\in X_1^{B}orx\in X_2^B,y\in {\bar{a}}_1\}\cup$$

$$\{[[a_1^j,z],a_1^k]:j\le k,z\in X_1^B\}\{[[x,a_1^j],y]:x\le y\in X_1^B\}{\bar{e}}_3{\overline{e^{\prime }}}_3$$

$$⟨U\bar{a}⟩=U{\otimes }_D⟨D\bar{a}⟩.$$

$$(M,⟨\bar{a}⟩){\equiv }_{L_{\infty ,\omega }}(N,⟨\bar{b}⟩).$$

Full text

1. Introduction

Zil’ber’s Conjecture is the following statement:

Let $T$ be an uncountably categorical theory in a countable language. If it is not one-based, then it is possible to interpret an infinite field in $T$.

The first counterexample was given by E.Hrushovski in 1988 (see [14]). It is a relational strongly minimal theory that does not even allow to interpret an infinite group.

In [4] a first group-counterexample is given . Using the classical results on groups of finite Morley rank it is easy to see that such a group-counterexample is essentially a simple group or a nilpotent group of finite exponent. The groups constructed in [4] are nilpotent of class 2 and of exponent $p>2$. In fact we worked with alternating bilinear maps. In the terminology of this paper they are 2-nilpotent graded Lie algebras over the field $\mathbb{F}(p)$ with p elements ($p>2$). If we try a similar construction in higher nilpotency classes additional difficulties arise. Here we will give 3-nilpotent counterexamples.

In his paper [15] on the fusion of two strongly minimal sets E. Hrushovski developed new ideas for such constructions as above. Together with A.Martin-Pizarro and M.Ziegler we used his ideas to obtain fields of prime characteristic of Morley rank 2 equipped with a definable additive subgroup of rank 1[10]. Furthermore we realised the fusion of two strongly minimal sets with DMP over a common vector space to obtain again a strongly minimal set[11]. Finally bad fields were construced by M.Hils, A.Martin-Pizarro, F.Wagner and me [12].

In [6] a common frame is built for the constuctions of the new uncountably categorical groups, the red fields, and the fusion over a vector space. There we have a starting theory $T$ that fulfills certain conditions. Notions like codes and difference sequences as in [10] and [11] are introduced and finally a collaps gives the desired theory of finite Morley rank.

In this paper we will follow this strategy of [6]. The main part (section 2 - 11) is devoted to the construction of an elementary theory of a 3-nilpotent graded Lie algebra over $\mathbb{F}(p)$, as a starting theory for the collaps. It has infinite Morley rank.

In section 2 we consider c-nilpotent graded Lie algebras $M=M_{1}\oplus\ldots\oplus M_{c}$ over a finite field $K$ in a suitable elementary language $L$. The $M_{i}$ are $K$-vector spaces. For $x\in M_{i}$ and $y\in M_{j}$ we have $[x,y]\in M_{i+j}$. A function $\delta$ over the set of all finite substructures $A$ of $M$ into the natural numbers is defined. For $c=2$ the definition is a minor deviation of the definition in [4]. Submodularity for the case $c=2$ is shown. Furthermore we define $A$ is strong in $M$, if $\delta(A)\leq\delta(B)$ for all finite $A\subseteq B\subseteq M$. For later amalgamation a class $\mathcal{K}_{c}$ is defined. In Lie algebras $M$ in $\mathcal{K}_{c}$ we have $0<\delta(A)$ for all substrucures $A\not=\langle 0\rangle$ of $M$.

In section 3 amalgamation is introduced. The existence of the free amalgam of c-nilpotent graded Lie algebras over a fixed field is shown in [7]. We study the structure of of the free amalgam in the case $c=3$.

In section 4 we describe the results for the case $c=2$ from [4] and [6].

From now on we work with 3-nilpotent Lie algebras. In section 5 a functor $\mathbb{F}$ from $\mathcal{K}_{c-1}$ into $\mathcal{K}_{c}$ is defined. For $A\in\mathcal{K}_{c}$ let $A^{*}=A/A_{c}$. If $B=\langle B_{1}\oplus\ldots\oplus B_{c-1}\rangle\in\mathcal{K}_{c}$ and $A^{*}\cong B^{*}$, then there is a homomorphism of $\mathbb{F}(A^{*})$ onto $B$. In the case of 3-nilpotent Lie algebras we show that a strong embedding of $B^{*}$ into $A^{*}$ in the sense of $\mathcal{K}_{2}$ implies that the natural homomorphism of $\mathbb{F}(B^{*})$ into $\mathbb{F}(A^{*})$ is an embedding. This is part of the main result (Theorem 5.5) in this section. It was proved in A.Amantini’s dissertation [1].

In section 6 we work again in $\mathcal{K}_{3}$. Submodularity of the $\delta$-function for substructures $A\in M$, where $A^{*}\leq M^{*}$, is shown. Then we prove the amalgamation property for $\mathcal{K}_{3}$ with strong embeddings.

Using this amalgamation we obtain in section 7 a countable strong Fraïssé-Hrushovski limit $\mathfrak{M}$ - the desired Lie algebra. We study the theory $T_{3}$ of $\mathfrak{M}$. $\mathfrak{M}$ is uniquely determined by richness: If $B$ is strong in $A$ and both are in $\mathcal{K}_{3}$, then a strong embedding $f$ of $B$ into $\mathfrak{M}$ can be extended to an strong embedding of $A$ into $\mathfrak{M}$. We give an axiomatization of $T_{3}$. In section 8 non-forking and canonical bases are investigated. $T_{3}$ is $\omega$-stable and CM-trivial.

Let $\mathfrak{C}$ be a monster model of $T_{3}$. In section 9 we define a pregeometry $cl$, using the $\delta$-function. Its domain is the union $R(\mathfrak{C})=\mathfrak{C}_{1}\cup C_{2}$ of the first two vector spaces of the graduation. The pregeometry is defined for all finite subspaces $A=\langle A_{1}A_{2}\rangle$. The smallest L-subspaces of this form are generated by a single element in $R(\mathfrak{C})=\mathfrak{C}_{1}\cup\mathfrak{C}_{2}$.

Our aim is to find a strong substructure $P^{\mu}(\mathfrak{C})$ of $\mathfrak{C}$, that has an uncountably categorical theory with the desired properties. For this we have to ensure that in the structure $P^{\mu}(\mathfrak{C})$ the geometrical closure is the algebraic closure. Therefore we study so-called minmal prealgebraic extension over substructures in section 10. We build formulas $\phi\in\mathcal{X}^{home}$ that describe these extensions. They are strongly minimal. To realise our aim above we define an expansion $\mathfrak{C}^{\mu}$ of $\mathfrak{C}$ by adding a new predicate $P^{\mu}$, such that in the substructure $P^{\mu}(\mathfrak{C}^{\mu})$ the number of solutions of the $\phi\in\mathcal{X}^{home}$ with parameters in $P^{\mu}(\mathfrak{C}^{\mu})$ is finite. The bound is given by a function $\mu$ on codes, that are modifications of the formulas in $\mathcal{X}^{home}$. There are uncountably many possible $\mu$-functions.

In [6] conditions are formulated that provide the existence of $\mathfrak{C}^{\mu}$. There we work with a pregeometry over a vector space, but here $R(\mathfrak{C})$ is the union of two vector spaces. Therefore we have to modify the conditions from [6]. In section 11 we show that $T_{3}$ satisfies these new conditions C(1)- C(7) for the collaps.

In the following sections we use our full knowledge of $T_{3}$ and not only these conditions. In section 12 we indroduce codes $\alpha$, that are modifications of the formulas in $\mathcal{X}^{home}$. Difference sequences are realizations of a spezial formulas $\psi_{\alpha}(\bar{x}_{0},\ldots,\bar{x}_{\lambda})$ that describes some properties of a sequence $\bar{a}_{0}-\bar{f},\ldots,\bar{a}_{\lambda}-\bar{f}$ ,where $\bar{a}_{0},\ldots,\bar{a}_{\lambda},\bar{f}$ is a Morley sequence of realizations of a code formula $\varphi_{\alpha}(\bar{x},b)$.

Then we introduce bounds for difference sequences in section 13. For this Lemma 13.1 provides an important combinatorial property of these sequences. It needs new ideas for the proof. We consider a class $\mathcal{K}^{\mu}$ of strong subalgebras of $\mathfrak{C}$, where a suitable function $\mu$ on the set of codes gives the desired bounds.

In section 14 we use the condition C(6) to amalgamate strong subalgebras from $\mathcal{K}^{\mu}$ inside $\mathfrak{C}$. We get a countable $\mathcal{K}^{\mu}$-rich strong subalgebra $P^{\mu}(\mathfrak{C})$ of $\mathfrak{C}$:

If $B\leq A$ are in $\mathcal{K}^{\mu}$ and if there is an strong embedding of $B$ into $P^{\mu}(\mathfrak{C})$, then we can extend the embedding to a strong embedding of $A$ into $P^{\mu}(\mathfrak{C})$.

In section 15 we extend our language by a predicate $P^{\mu}$ for this subalgebra. The new language is denoted by $L^{\mu}$. A $L^{\mu}$-structure $(M,P^{\mu}(M))$ is a $2\times$rich $L^{\mu}$-structure, if $M$ is a rich model of $T_{3}$, $P^{\mu}(M)\leq M$ is $\mathcal{K}^{\mu}$-rich, and the geometrical dimension of $M$ over $P^{\mu}(M)$ is infinite. These structures have a complete theory $T^{\mu}_{3}$, with a monster model $\mathfrak{C}^{\mu}$.

In section 16 $T^{\mu}_{3}$ is axiomatized. We get $\omega$-stability. Let $\Gamma(\mathfrak{C}^{\mu})$ be the $L$-substructure with domain $P^{\mu}$. It is the desired new uncountablly categorical graded 3-nilpotent Lie algebra over a finite field. It will be considered in section 17. In this substructure $cl$ is part of the algebraic clousure. $\Gamma(\mathfrak{C}^{\mu})$ is stably embedded in $\mathfrak{C}^{\mu}$. In $\Gamma(\mathfrak{C}^{\mu})$ the predicate $R$ has Morley rank 1 and Morley degree 2. The geometry of the algebraic closure is not locally modular, but the theory is CM-trival. It is not possible to interpret an infinite field. The Morley rank of this structure is 3.

In section 18 we get group counterexamples by interpretation. We use the Baker-Campbell-Hausdorff-formula. Here the sum is finite. These groups are bi-interpretable with the Lie algebras of $T_{3}^{\mu}$ without graduation. Their theories are uncountably categorical. They are not one-based, CM-trivial, and do not allow the interpretation of a field.

2. Definition of $\delta$

We consider c-nilpotent graded Lie algebras $M$ over a finite field $K$ in a language $L$. The non-logical symbols of $L$ are the following: There are $+$, $-$,[math] and unary functions for the scalar multiplication with the elements of $K$ to describe the underlying vector space. Furthermore we have unary predicates $U_{i}$ ($1\leq i\leq c$) for the graduation. That means $M=\oplus_{1\leq i\leq c}M_{i}$, where $M_{i}=U_{i}(M)$. Projections $pr_{i}$ are needed to ensure that substructures are again graded. $[x,y]$ is the symbol for the Lie - multiplication. In $M$ $U_{i}(a)$ and $U_{j}(b)$ implies $U_{i+j}([a,b])$. $\langle X\rangle$ denotes the $L$-substructure of $M$ generated by $X\subset M$. $\langle X\rangle^{lin}$ denotes the linear hull of $X$. If $a\in U_{i}(M)$, then we say that $a$ is homogeneous of degree $i$. A subset $X$ is homogeneous, if all its elements are homogenous. If $a\in M$ then $a=\sum_{1\leq i\leq c}r_{i}a_{i}$, where $a_{i}\in U_{i}(M)=M_{i}$. The degree of $a$ is the smallest $i$, such that $r_{i}\not=0$. Finally $\langle\langle X\rangle\rangle$ is the ideal generated by $X$ in $M$. $ldim$ is used to denote the linear dimension.

$A,B,C,D$ denote finite and $M,N$ arbitrary c-nilpotent graded Lie algebras over $K$. For every finite $A$ we define an integer $\delta(A)$ that is uniquely determined by the isomorphism type of $A$. Note that we do not assume in general that $A=\langle A_{1}\rangle$.

We use the Theorem of Sirsov-Witt: Every subalgebra of a free Lie algebra over a field is free. In [3] a key lemma for the proof of this Theorem is formulated, that is useful from a model-theoretic point of view. Here we develop the results in a context with graduation.

If $F(X)$ is the the free Lie algebra freely generated by $X$, then $F(X)$ becomes a free graded Lie algebra, if we define $U_{i}(F(X))$ to be the vector space generated by all momomials over $X$ of degree $i$. $U_{1}(F(X))$ is the linear hull of $X$. Projections are defined in the usual way. Then we have

[TABLE]

where $Z_{1}(M)\subseteq\ldots\subseteq Z_{c}(M)$ is the upper central series and $\Gamma_{1}(M)\supseteq\ldots\supseteq\Gamma_{c}(M)$ the lower central series.

Definition 2.1**.**

Let $Y$ be a homogeneous subset of $M$.

1. (1)

$Y$ is an (o)-system in $M$, if for all $i$ ($1\leq i\leq c$) $U_{i}(Y)$ is linearly independent over $\langle U_{1}(Y),\ldots,U_{i-1}(Y)\rangle_{i}$. 2. (2)

$Y$ generates $M$ freely, if $Y$ is an o-system in $M$, that generates $M$ and for every c-nilpotent graded Lie algebra $N$ over $K$ every map $f$ of $Y$ into $N$ with $U_{i}(y)$ implies $U_{i}(f(y))$ can be extended to an $L$-homomorphism of $M$ into $N$. We call such a Lie algebra $M$ the free algebra $F(Y)$.

The fact that $Y$ generates $M$ freely can be expressed by linear independence of basic commutators.

In [3] the following Theorem is taken from the proof of the Sirsov-Witt-Theorem and applied for the following corollaries. Here we formulate it with graduation.

Theorem 2.2**.**

In a free graded c-nilpotent Lie Algebra $F(X)$ every (o)-system $Y$ freely generates $\langle Y\rangle$.

Corollary 2.3**.**

Every element of $F(Y)$ has a unique presentation as a linear combination of basic commutators over $Y$.

Definition 2.4**.**

If $X=\cup_{1\leq i\leq c}X_{i}$ where $X_{i}\subseteq U_{i}(M)$ is a generating o-system for M, then we define $o-dim_{i}(M)=\mid X_{i}\mid$. $o-dim(M)=\sum_{1\leq i\leq c}o-dim_{i}(M)$.

It is easily seen, that $o-dim_{i}(M)$ is independent from the choice of $X$.

Corollary 2.5**.**

If $M$ and $N$ are free and $o-dim_{i}(M)=o-dim_{i}(N)$ for $1\leq i\leq c$, then $M\simeq N$.

We choose a generating o-system $W$ for some c-nilpotent graded Lie algebra $M$. Let $F(X)$ be the free graded c-nilpotent Lie algebra over $K$ freely generated by $X$, where $X$ be an o-system with $\mid X_{i}\mid=\mid W_{i}\mid$ for $1\leq i\leq c$. Then for every map of $X_{i}$ onto $W_{i}$ for all $i$ we get a homorphism $f$ of $F(X)$ onto $M$. Hence $M\simeq F(X)/ker(f)$.

Definition 2.6**.**

We call $F(X),ker(f)$ as above a canonical pair for $M$. As we will see below it is unique up to automorphisms of $F(X)$.

If $g$ is any homorphism of $F(X)$ onto $M$, then there is a generating o-system $X^{0}$ in $F(X)$ such that $g(X^{0}_{i})=W_{i}$. By Corollary 2.5 there is an automorphism $\alpha$ with $\alpha(X_{i})=X^{0}_{i}$. Hence $f=g\alpha$ and $ker(g)$ is an automorphic image of $ker(f)$. It follows that the following definition is independent of $f$.

Definition 2.7**.**

An ideal basis for $M\simeq F(X)/ker(f)$ is a homogeneous subset $Y$ of $ker(f)$, such that $Y_{i}$ generates $ker(f)\cap F(X)_{i}$ modulo $\langle\langle Y_{2},\ldots,Y_{i-1}\rangle\rangle\cap F(X)_{i}$ for $2\leq i\leq c$. For finite $M$ we define $ideal-dim_{i}(M)=\mid Y_{i}\mid$ and

[TABLE]

and $\delta(M)=\delta_{c}(M)$.

$o-dim_{i}(M)$, $ideal-dim_{i}$, and $\delta_{i}(M)$ are invariants of $M$. They depend only on the isomorphism-typ of $M$. In the approach in [4] and in [6] for $c=2$ we consider only substructures $A$ with $A=\langle A_{1}\rangle$. In that context the proof of the next lemma is easier.

Lemma 2.8**.**

Submodularity holds for $\delta_{2}$. Let $M$ be a 2-nilpotent graded Lie algebra. Let $A$ and $C$ be subalgebras of $M$. Then

[TABLE]

Proof.

Define $A_{i}\cap C_{i}=B_{i}$. Let $X_{1}=X_{1}(AC)=X_{1}(C)X_{1}(B)X_{1}(A)$ be a vector basis for $\langle CA\rangle_{1}$ such that:

**: **

$X_{1}(B)$ is a vector basis of $B_{1}$.

**: **

$X_{1}(B)X_{1}(C)$ is a vector basis of $C_{1}$.

**: **

$X_{1}(B)X_{1}(A)$ is a vector basis of $A_{1}$.

We define:

**: **

$X_{2}^{0}(B)$ is a vector basis for $\langle C_{1}\rangle_{2}\cap\langle A_{1}\rangle_{2}$ over $\langle B_{1}\rangle_{2}$.

**: **

$X_{2}^{A}(B)$ is a vector basis for $\langle A_{1}\rangle_{2}\cap C_{2}$ over $\langle B_{1}\rangle_{2}+\langle X_{2}^{0}(B)\rangle$.

**: **

$X_{2}^{C}(B)$ is a vector basis for $\langle C_{1}\rangle_{2}\cap A_{2}$ over $\langle B_{1}\rangle_{2}+\langle X_{2}^{0}(B)\rangle$.

**: **

$X_{2}^{1}(B)$ is a vector basis for $B_{2}\cap\langle X_{1}\rangle_{2}$ over $\langle B_{1}\rangle_{2}+\langle X_{2}^{0}(B)X_{2}^{A}(B)X_{2}^{C}(B)\rangle$.

**: **

$X_{2}^{2}(B)$ is a vector basis of $B_{2}$ over $\langle X_{1}\rangle_{2}$.

Note that $X_{2}^{A}(B)\cup X_{2}^{C}(B)$ is linearly independent over $\langle X_{1}(B)\rangle_{2}+\langle X_{2}^{0}(B)\rangle_{2}$. Now $B_{2}$ has the following vector basis $X_{2}(B)$ over $\langle B_{1}\rangle$:

[TABLE]

Let $X_{2}^{1}(A)$ and $X_{2}^{1}(C)$ be choosen such that

**: **

$X_{2}^{C}(B)X_{2}^{1}(B)X_{2}^{1}(A)$ is a vector basis of $A_{2}\cap\langle A_{1}C_{1}\rangle_{2}$ over $\langle A_{1}\rangle_{2}$.

**: **

$X_{2}^{A}(B)X_{2}^{1}(B)X_{2}^{1}(C)$ is a vector basis of $C_{2}\cap\langle A_{1}C_{1}\rangle_{2}$ over $\langle C_{1}\rangle_{2}$.

Choose $X_{2}^{2}(A)$ and $X_{2}^{2}(C)$ such that

**: **

$X_{2}(A)=X_{2}^{C}(B)X_{2}^{1}(B)X_{2}^{2}(B)X_{2}^{1}(A)X_{2}^{2}(A)$ is a vector basis of $A_{2}$ over $\langle A_{1}\rangle_{2}$.

**: **

$X_{2}(C)=X_{2}^{A}(B)X_{2}^{1}(B)X_{2}^{2}(B)X_{2}^{1}(C)X_{2}^{2}(C)$ is a vector basis of $C_{2}$ over $\langle C_{1}\rangle_{2}$.

Then $X_{2}(AC)=X_{2}^{2}(B)X_{2}^{2}(A)X_{2}^{2}(C)$ is a vector basis of $\langle AC\rangle_{2}$ over $\langle A_{1}C_{1}\rangle_{2}$. Hence $X_{1}(AC)X_{2}(AC)$ is a generating o-system of $\langle AC\rangle$. Then

[TABLE]

where $I$ is a subspace of $F(X_{1}(AC)_{2}$. We get

1. (1)

$\delta_{2}(\langle AC\rangle=\mid X_{1}(AC)\mid+\mid X_{2}(AC)\mid-ldim(I)$ 2. (2)

$\delta_{2}(A)=\mid X_{1}(B)\mid+\mid X_{1}(A)\mid+\mid X_{2}(A)\mid-ldim(I\cap F(X_{1}(B)X_{1}(A))$. 3. (3)

$\delta_{2}(C)=\mid X_{1}(B)\mid+\mid X_{1}(C)\mid+\mid X_{2}(C)\mid-ldim(I\cap F(X_{1}(B)X_{1}(C))$. 4. (4)

$\delta_{2}(B)=\mid X_{1}(B)\mid+\mid X_{2}(B)\mid-ldim(I\cap F(X_{1}(B))$. 5. (5)

$o-dim_{1}(A)+o-dim_{1}(C)-o-dim_{1}(B)=o-dim_{1}(AC)$. 6. (6)

$o-dim_{2}(A)+o-dim_{2}(C)-o-dim_{2}(B)=$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE] 7. (7)

$\delta_{2}(A)+\delta_{2}(C)-\delta_{2}(B)=\delta_{2}(AC)+\mid X_{2}^{1}(B)X_{2}^{1}(A)X_{2}^{1}(C)\mid-\mid X_{2}^{0}(B)\mid+$

[TABLE]

Let $I^{0}$ be $(I\cap F(X_{1}(B)X_{1}(A)))\oplus_{I\cap F(X_{1}(B))}(I\cap F(X_{1}(B)X_{1}(C))$. Then $I^{0}$ is a subspace of $I$ and $I$ contains $\mid X_{2}^{0}(B)\mid$ - many elements $\psi_{i}=\psi_{i}(A)+\psi_{i}(C)$ with $\psi_{i}(A)\in F(X_{1}(B)X_{1}(A))$ and $\psi_{i}(C)\in F(X_{1}(B)X_{1}(C))$, that are linearly independent over $I^{0}$. Hence

[TABLE]

∎

We summeraize the results of the computation above.

Corollary 2.9**.**

*As above $B=A\cap C$. We define:

$r_{B}=ldim((\langle A_{1}C_{1}\rangle_{2}\cap B_{2})/\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$,

$r_{A}=ldim((\langle A_{1}C_{1}\rangle_{2}\cap A_{2})/\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$,

$r_{C}=ldim((\langle A_{1}C_{1}\rangle_{2}\cap C_{2})/\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$, and

if $\langle A_{1}C_{1}\rangle=F(A_{1}C_{1})/I$, then $s=ldim(I/(I\cap(F(A_{1}))+F(C_{1}))$. Then*

1. (1)

$\delta_{2}\langle AC\rangle)-\delta_{2}(C)=\delta_{2}(A)-\delta_{2}(B)-(r_{A}+r_{C}-r_{B}+s).$** 2. (2)

$\delta_{2}(\langle AC\rangle)=\delta_{2}(A)+\delta_{2}(C)-\delta_{2}(A\cap C)$* if and only if *

$r_{A}=r_{C}=r_{B}=s=0$* if and only if*

$A_{2}$* and $C_{2}$ do not contain elements in $\langle A_{1}C_{1}\rangle\setminus\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$,*

and if $\langle A_{1}C_{1}\rangle=F(A_{1}C_{1})/I$, then $I=(I\cap(F(A_{1}))+F(C_{1}))$.

Definition 2.10**.**

Let $M$ be a graded c-nilpotent Lie algebra over a finite field $K$. Let $A$ be a substructure of $M$.

1. (1)

For $2\leq i\leq c$ $A$ is i-strong in $M$ (short $A\leq_{i}M$), if for all $2\leq j\leq i$ and all $A\subseteq C\subseteq M$ we have $\delta_{j}(A)\leq\delta_{j}(C)$. 2. (2)

We write $A\leq M$, if $A\leq_{c}M$. In this case we say $A$ is strong in $M$. We use $A\leq_{1}M$ for $A\subseteq M$.

For the graded Lie algebras $M\in\mathcal{K}_{3}$, that we will consider later, we will have:

if $\delta_{3}(A)\leq\delta_{3}(C)$ for all finite $C$ with $A\subseteq C\subseteq M$, then

$\delta_{2}(A)\leq\delta_{2}(C)$ for all finite $C$ with $A\subseteq C\subseteq M$.

Lemma 2.11**.**

Let $M$ be a graded c-nilpotent Lie algebra over a finite field $K$. We assume for all (i-1)-strong substructures of $M$ submodularity for $\delta_{i}$ is true. Furthermore for all substructures $A,C,E$ of $M$,where $A,C$ are finite, we have:

1. (0)

If $A\subseteq E$, then there is some $A\subseteq A^{\prime}\leq_{i-1}E$ and $\delta_{i}(A^{\prime})\leq\delta_{i}(A)$. 2. (2,i-1)

If $A\leq_{i-1}C\leq_{i-1}M$, then $A\leq_{i-1}M$. 3. (3,i-1)

If $A,C\leq_{i-1}M$, then $A\cap C\leq_{i-1}M$.

Then for all substructures $A,C,E$ of $M$, where $A,C$ finite the following is true:

1. (1)

If $C\leq_{i}M$ and $E\leq_{i-1}M$, then $E\cap C\leq_{i}E$. 2. (2)

If $A\leq_{i}C\leq_{i}M$, then $A\leq_{i}M$. 3. (3)

If $A,C\leq_{i}M$, then $A\cap C\leq_{i}M$.

Proof.

ad 1) By assumption $C,E\leq_{i-1}M$. Choose any $D$ such that $E\cap C\subseteq D\subseteq E$. By (3,i-1) we have $E\cap C\leq_{i-1}M$. By (0) we have some $D^{\prime}$, such that $D\subseteq D^{\prime}\leq_{i-1}E$ and $\delta_{i}(D^{\prime})\leq\delta_{i}(D)$. By (2,i-1) we get $D^{\prime}\leq_{i-1}M$. By submodularity for (i-1)-strong substructures of $M$ we have

[TABLE]

Note $D^{\prime}\cap C=D\cap C=E\cap C$. Hence

[TABLE]

since $C\leq_{i}M$. It follows $\delta_{i}(E\cap C)\leq\delta_{i}(D^{\prime})\leq\delta_{i}(D)$ for all $D$ between $E\cap C$ and $E$.

ad 2) By (2,i-1) we have $A\leq_{i-1}M$. Also $C\leq_{i-1}M$. Consider $A\subseteq E\subseteq M$. If $E\subseteq C$ or $C\subseteq E$, then $\delta_{i}(A)\leq\delta_{i}(E)$ or $\delta_{i}(A)\leq\delta_{i}(C)\leq\delta_{i}(E)$ respectively. Otherwise we apply (0) and obtain $E\subseteq E^{\prime}\leq_{i-1}M$ with $\delta_{i}(E^{\prime})\leq\delta_{i}(E)$. By submodularity of $\delta_{i}$ for (i-1)-strong substructures

[TABLE]

By $C\leq_{i}M$ and since $\delta_{i}(A)\leq\delta_{i}(E^{\prime}\cap C)$

[TABLE]

Hence $\delta_{i}(A)\leq\delta_{i}(E^{\prime})\leq\delta_{i}(E)$.

ad 3) By 1) we have $A\cap C\leq_{i}A\leq_{i}M$ and by 2) $A\cap C\leq_{i}M$.

∎

Note 1-strong substructures are substructures. Submodularity for $\delta_{2}$ for substructures is shown in Lemma 2.8. Hence

Corollary 2.12**.**

For all substructures $A,C,E$ of $M$, where $A,C$ are finte the following is true:

1. (1)

If $C\leq_{2}M$ and $E\subseteq M$, then $E\cap C\leq_{2}E$. 2. (2)

If $A\leq_{2}C\leq_{2}M$, then $A\leq_{2}M$. 3. (3)

If $A,C\leq_{2}M$, then $A\cap C\leq_{2}M$.

Let $M$ be $F(X)/\langle\langle Y\rangle\rangle$, where $X$ is a generating o-system with $o-dim_{i}(M)=\\ \mid X_{i}\mid$ and $Y$ is an ideal basis of $ker(f)$, where $f$ is as above $f:F(X)\rightarrow M$, $f$ is onto, and $f(X)$ is a generating o-system of $M$.

Let $M^{\prime}$ be $M/M_{i+1}\oplus\ldots\oplus M_{c}$ and $\tau$ the canonical homomorphism of $M$ onto $M^{\prime}$. Then $\tau$ is one-to-one on $M_{1}\oplus\ldots\oplus M_{i}$ and $ker(\tau)=M_{i+1}\oplus\ldots\oplus M_{c}$.

Lemma 2.13**.**

Assume $M\subseteq N$ and $F(X),ker(f)$ is a canonical pair for $N$ as above. Then there is a subalgebra $H$ of $F(X)$, such that $H,H\cap ker(f)$ is a canonical pair for $M$.

Note that $H$ is free by Theorem2.2.

Proof.

Choose an generating o-system $Z$ for $M$ and let $Z^{0}$ be a $f$-preimage of $Z$ in $F(X)$. Let $H$ be the substructure of $F(X)$ generated by $Z^{0}$. ∎

Definition 2.14**.**

If $M\subseteq N$, then $X=\cup_{1\leq i\leq c}X_{i}$ is an o-system for $N$ over $M$,if $U_{i}(X_{i})$, $X_{1}$ is minimal with $\langle M_{1}X_{1}\rangle_{1}=N_{1}$ and $X_{i}$ is minimal with $\langle MX_{1}\ldots X_{i}\rangle_{i}=N_{i}$. Then $o-dim(N/M)=\sum_{1\leq i\leq c}\mid X_{i}\mid$ and $o-dim_{i}(N/M)=\mid X_{i}\mid$.

In the context of Lemma 2.13 $Y=\cup_{2\leq i\leq c}Y_{i}$ is an ideal basis of $N$ over $M$, if $Y_{i}$ is a vector space basis of $F(X)_{i}\cap ker(f)$ over $F(X)_{i}\cap(\langle\langle(ker(f)\cap H),Y_{2},\ldots,Y_{i-1}\rangle\rangle)$. Then $ideal-dim_{i}(N/M)=\mid Y_{i}\mid$ and $ideal-dim(N/M)=\sum_{2\leq i\leq c}ideal-dim_{i}(N/M)$.

We extend Definition 2.10

Definition 2.15**.**

Let $M$ be a graded c-nilpotent Lie algebra over a finite field. Let $A$ be a finite substructure.

1. (3)

$A$ is strong in $M$ restricted by $k$ (short $A\leq^{k}M$), if for all $A\subseteq C\subseteq M$ with $\sum_{1\leq i\leq c-1}o-dim_{i}(C/A)\leq k$, we have $\delta(A)\leq\delta(C)$.

Note that $\delta(C)=\delta(\langle C_{1},\ldots,C_{c-1}\rangle)+o-dim_{c}(C)$.

Definition 2.16**.**

Let $\mathcal{K}_{c}$ be the class of all graded c-nilpotent Lie algebras $M$ over a fixed finite field $K$ considered as $L$ - structures. such that the following is true:

For $A\subseteq M$ we have $min\{o-dim(A),2\}\leq\delta(A)$

Lemma 2.17**.**

Assume $M\in\mathcal{K}_{c}$. Given $i+j\leq c$ in $M$ holds

[TABLE]

We say that there are no homogeneous zero-divisors.

Proof.

Assume $U_{i}(a)$, $U_{j}(b)$, $i+j\leq c$ and $i<j$ or $i=j$ and $a$ and $b$ are linearly independent . By assumption $o-dim(\langle a,b\rangle)=\delta(\langle a,b\rangle)=2$. Hence $[a,b]\not=0$. ∎

Definition 2.18**.**

Let $\mathcal{H}_{c}$ be the subclass of all $M\in\mathcal{K}_{c}$ with $M=\langle M_{1},\ldots,M_{c-1}\rangle$. $\mathcal{K}_{c}^{fin}$ and $\mathcal{H}_{c}^{fin}$ are the subclasses of finite sutructures.

$\mathcal{K}_{c}$ and $\mathcal{K}_{c}^{fin}$ are closed under substructures.

Corollary 2.19**.**

1. (1)

Let $M$ be a graded 2-nilpotent Lie algebra ovr a finite field. $M\in\mathcal{K}_{2}$ if and only if for every $A=\langle A_{1}\rangle\subseteq M$ with $lin-dim(A_{1})\geq 2$ we have $\delta_{2}(A)\geq 2$. 2. (2)

Let $M$ be a graded 3-nilpotent Lie algebra over a finite field. $M\in\mathcal{K}_{3}$ if and only if for every $A=\langle A_{1}A_{2}\rangle\subseteq M$ with $o-dim(A)\geq 2$ we have $\delta_{3}(A)\geq 2$.

3. Free amalgamation

Let $\mathcal{K}$ be a class of structures as e.g. the class of all c-nilpotent graded Lie algebras over a fixed field, $\mathcal{K}_{c}$, or $\mathcal{K}_{c}^{fin}$.

Definition 3.1**.**

We define the amalgamation and the free amalgamation for $\mathcal{K}$.

**AP: **

Amalgamation Property Assume $g_{0}:B\to A$ and $g_{1}:B\to C$ are embeddings for $A,B,C\in\mathcal{K}$. Then there are some $D$ in $\mathcal{K}$ and embeddings $f_{0}:A\to D$ and $f_{1}:C\to D$ such that $f_{0}\circ g_{0}=f_{1}\circ g_{1}$ for $B$.

** APS: **

We have the strong amalgamation property for $\mathcal{K}$ if in AP $f_{0}(A)\cap f_{1}(C)=f_{0}\circ g_{0}(B)=f_{1}\circ g_{1}(B)$ holds.

**Free Amalgam: **

Let $A,B,C,D\in\mathcal{K}$ and assume that $B$ is a common substructure of $A$ and $C$. If $D$ is generated by $A$ and $C$ with $A\cap C=B$, then $D$ is the free amalgam of $A$ and $C$ over $B$ (short $D=A\otimes_{B}C$) in $\mathcal{K}$, if for all homomorphisms $f:A\to E$ and $g:C\to E$ into some $E\in\mathcal{K}$ with $f(b)=g(b)$ for $b\in B$ there is a homomorphism $h:D\to E$ that extends $f$ and $g$.

**Closed: **

$\mathcal{K}$ is closed under free amalgamation, if for $A,B,C\in\mathcal{K}$ and embeddings $g_{0}:B\to A$ and $g_{1}:B\to C$, there exists a free amalgam $A^{\prime}\otimes_{B^{\prime}}C^{\prime}$ in $\mathcal{K}$ and isomorphisms $f_{0}:A\to A^{\prime}$ and $f_{1}:C\to C^{\prime}$ , such that $f_{0}\circ g_{0}(b)=f_{1}\circ g_{1}(b)$ for $b\in B$ maps $B$ onto $B^{\prime}$ .

The free amalgam is a strong amalgam by definition. The homomorphism $h:D\to E$ in the definition is unique, since $D$ is generated by $A$ and $C$. Note that $A\otimes_{B}C$ is uniquely determined up to isomorphisms, if it exists.

We define as in [7]:

Definition 3.2**.**

For subsets $A,B,C$ in a structure $M$ we define

[TABLE]

if and only if

[TABLE]

.

Let $L$ be countable. K.Tent and M.Ziegler defined a stationary independence relation for the investigation of automorphism groups in [20]. We consider finite subsets $A,B,C,D$ of a $L$-structure $M$.

Definition 3.3**.**

A relation A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C for finite subsets of $M$ is called a stationary independence relation in $M$ if it fulfils the following properties.

**Inv: **

Invariance A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C depends only on the elementary type of $A,B,C$.

**Mon: **

Monotonicity A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}CD implies A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C and A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{BC}D.

**Trans: **

Transitivity A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C and A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{BC}D imply A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}CD.

**Sym: **

Symmetry A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C if and only if C\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}A.

**Ex: **

Existence For $A,B,C$ there is some $A^{\prime}$ in $M$ such that $\operatorname{tp}(A/B)=\operatorname{tp}(A^{\prime}/B)$ and A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C.

**Stat: **

Stationarity If $\operatorname{tp}(A/B)=\operatorname{tp}(A^{\prime}/B)$, A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C, and A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C, then $\operatorname{tp}(A/BC)=\operatorname{tp}(A^{\prime}/BC)$.

In [7] the following is shown. Note that the graduation is essential for the amalgamation.

Theorem 3.4**.**

The class of c-nilpotent graded Lie algebras over a field $K$ is closed under free amalgamation.

Theorem 3.5**.**

If $K$ is a finite field, then the Fraïssé limit $M_{0}$ of all finitely generated c-nilpotent graded Lie algebras exists. If $\mathfrak{C}^{uni}$ is a monster model of $Th(M_{0})$, then the free amalgam defines a stationary independence relation \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}in $\mathfrak{C}^{uni}$.

Then every graded Lie algebra we consider is a substructure in $\mathfrak{C}^{uni}$, and if $B\subseteq A$, then every embedding of $B$ in $\mathfrak{C}^{uni}$ can be extended to $A$. Furthermore $tp(A/B)$ is completely determined by it’s quantifier-free part. That means we can use the properties of a stationary independence relation for the free amalgam in the class of c-nilpotent graded Lie algebras over $K$.

Definition 3.6**.**

Let $M$ be a graded c-nilpotent Lie algebra. By a vector space basis of $M$ we mean the union of the vector space bases $X_{i}$ of all subspaces $M_{i}$ of the graduation.

In the proof of the Theorem 3.4 we prove the following Major Case. The final construction of the free amalgam is an iteration of it.

**Major Case: **

Assume $A=\langle Ba\rangle$ and $C=\langle Be\rangle$ with $U_{i}(a)$ , $U_{j}(e)$, and $i,j<c$. Furthermore we have $[a,b]\in B$ and $[e,b]\in B$ for $b\in B$. Then the free amalgam $D$ of $A$ and $C$ over $B$ exists. Let $X^{B}$ be a homogeneous vector space basis of $B$, Let $Y$ be a vector space basis of the free graded c-nilpotent Lie algebra freely generated by $a$ and $e$. We can assume that $Y$ is a set of basic monomials over $a,e$. Then $X^{B}Y$ is a vector space basis of $D$ and the Lie multiplication is inductively defined by the Jacobi identity, using $[a,b]\in B$ and $[e,b]\in B$ for $b\in B$.

Corollary 3.7**.**

Let $C,B,A$ be finite graded 2-nilpotent Lie algebras over a field. Let $X^{B}=X_{1}^{B}X_{2}^{B}$ be a vector space basis of $B$, $X^{A}=X_{1}^{A}X_{2}^{A}$ a vector space basis of $A$ over $X^{B}$, and $X^{C}=X_{1}^{C}X_{2}^{C}$ a vector space basis of $C$ over $X^{B}$. Then

[TABLE]

is a vector space basis of $A\otimes_{B}C$.

Using Corollary 2.9 we obtain:

Corollary 3.8**.**

Let $M$ be a 2-nilpotent Lie algebra and let $B\subseteq A,B\subseteq C$ be subalgebras of $M$ with $A\cap C=B$. Then the following are equivalent:

1. (1)

$\langle AC\rangle=A\otimes_{B}C$. 2. (2)

$\delta(\langle AC\rangle)-\delta(C)=\delta(A)-\delta(B)$. 3. (3)

There are no elements of $\langle A_{1}C_{1}\rangle\setminus\langle A_{1}\rangle+\langle C_{1}\rangle$ in $A_{2}$ or in $C_{2}$. Furthermore if $\langle A_{1}C_{1}\rangle=F(A_{1}C_{1})/I$, then $I\subseteq\langle A_{1}\rangle^{F(A_{1}C_{1})}+\langle C_{1}\rangle^{F(A_{1}C_{1})}$.

Proof.

(2) and (3) are equivalent by Corollary 2.9. If $\langle AC\rangle=A\otimes_{B}C$, then by Corollary 3.7 we obtain (3). Finally we show that (3) implies (1). Let $f$ be a homomorphism of $A$ into $E$ and $g$ of $C$ into $E$ such that $f$ and $g$ coincide over $B=A\cap C$. As above let $X^{B}=X_{1}^{B}X_{2}^{B}$ be a vector space basis of $B$, $X^{A}=X_{1}^{A}X_{2}^{A}$ a vector space basis of $A$ over $X^{B}$, and $X^{C}=X_{1}^{C}X_{2}^{C}$ a vector space basis of $C$ over $X^{B}$. Then by the conditions of (3) $\{[x,y]:x\in X_{1}^{A},y\in X_{1}^{C}\}$ is linearly independent over $\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$. Therefore and by the second condition the homomorphism $h$ of $F(X_{1}(B)X_{1}(A)X_{!}(C))$ into $E$, given by $f$ and $g$, induces a homomorphism of $\langle A_{1}C_{1}\rangle$ into $E$. Then it is no problem to extend this to $\langle AC\rangle$. ∎

Corollary 3.9**.**

Let $C,B,A$ be finite graded 3-nilpotent Lie algebras over a field with $A\cap C=B$. Then the following are equivalent:

1. (1)

$\langle AC\rangle=A\otimes_{B}C$. 2. (2)

Let $X^{B}=X_{1}^{B}X_{2}^{B}X_{3}^{B}$ be a vector space basis of $B$, $X^{A}=X_{1}^{A}X_{2}^{A}X_{3}^{A}$ a vector space basis of $A$ over $X^{B}$, and $X^{C}=X_{1}^{C}X_{2}^{C}X_{3}^{C}$ a vector space basis of $C$ over $X^{B}$. Assume $X_{1}^{A}$ and $X_{1}^{C}$ are ordered. Then the following is a vector space basis of $\langle AC\rangle$:

[TABLE]

[TABLE]

[TABLE]

Proof.

It is easily seen, that there exists a graded Lie-algebra $M$ with the vector space basis (2). We have only to define $[x,y]$ in a canonical way and then we have to show that the Jacobi identity is true. Using the definiton we get that $M\cong A\otimes_{B}C$. ∎

The following property of the free amalgam of 3-nilpotent graded Lie-algebras over a field will be later useful:

Theorem 3.10**.**

In a 3-nilpotent graded Lie algebra over a field we consider $U\subseteq V$, $\bar{a}$ a sequence of elements, $\langle U\bar{a}\rangle\cap V=U$, $B\subseteq U$, and $C\subseteq V$ such that

1. (1)

$\langle V\bar{a}\rangle=V\otimes_{C}\langle C\bar{a}\rangle$, 2. (2)

$\langle U\bar{a}\rangle=U\otimes_{B}\langle B\bar{a}\rangle$.

*Then $\langle U\bar{a}\rangle=U\otimes_{D}\langle D\bar{a}\rangle$, where $D=C\cap B$.

The same is true for $c=2$.*

Proof.

W.l.o.g. $\bar{a}$ is a sequence of homogenous elements, since there are projections in the language. By the assumptions a set of monomials of homegeneous elements from $\langle U\bar{a}\rangle$ that is linearly independent over $U$ is linearly indepentent over $V$. Then $\langle\bar{a}\rangle^{lin}\cap U=\langle\bar{a}\rangle^{lin}\cap V$. Hence we can assume w.l.o.g., that $\bar{a}$ is linearly independent over $U$ and $V$. Furthermore we suppose w.l.o.g., that $\bar{a}_{2}$ is linearly independent over $V_{2}+\langle\bar{a}_{1}\rangle_{2}$ and $\bar{a}_{3}$ is linearly independent over $\langle\bar{a}_{1}\bar{a}_{2}\rangle_{3}+V_{3}$.

Assume w.l.o.g. $\bar{a}=\bar{e}\bar{e^{\prime}}\bar{e^{\prime\prime}}$ with the following properties:

1. (3)

$\bar{e}$ is a generating o-system for $\langle V\bar{a}\rangle$ over $V$.

Then $\bar{e_{1}}=\bar{a_{1}}$ and $\bar{e}$ is an o-system over $U$. 2. (4)

$\bar{e}\bar{e^{\prime}}$ is a generating o-system of $\langle U\bar{a}\rangle$ over $U$.

Note that $\bar{e^{\prime}}_{1}$ and $\bar{e^{\prime\prime}}_{1}$ are empty.

Assume $e\in\bar{e^{\prime}}_{2}\bar{e^{\prime\prime}}_{2}$. Then w.l.o.g.

[TABLE]

where $v_{2}\in V_{2}$ and $v_{1}^{j}\in V_{1}$.

By (1) and Corollary 3.9 we get $v_{1}^{j}\in C_{1}$ and $v_{2}=e-\sum_{j}[a_{1}^{j},v_{1}^{j}]\in\langle C\bar{a}\rangle\cap V=C$.

If $e\in\bar{e^{\prime\prime}}_{2}$, then $e\in\langle U\bar{a}_{1}\rangle$. Then by (2) and Corollary 3.9 $v_{1}^{j}\in B_{1}$ and $v_{2}\in\langle B\bar{a}\rangle\cap U=B$. By the considerations above $v_{1}^{j}\in D_{1}$ and $v_{2}\in D_{2}$.

We can rewrite (1) as $\langle V\bar{a}\rangle=V\otimes_{C}\langle C\bar{a}_{1}\bar{e}_{2}\bar{a}_{3}\rangle$

and (2) as $\langle U\bar{a}\rangle=U\otimes_{B}\langle B\bar{a}_{1}\bar{e}_{2}\bar{e^{\prime}}_{2}\bar{a}_{3}\rangle$.

Let $X_{i}^{D}$ be a vector space basis of $D_{i}$. We can extend $X_{i}^{D}$ by $X_{i}^{B}$, such that $X_{i}^{D}X_{i}^{B}$ is a vector space basis of $B_{i}$. Analogously we get $X_{i}^{C\cap U}$, such that $X_{i}^{D}X_{i}^{C\cap U}$ is a vector space basis of $C_{i}\cap U_{i}$. Let $X_{i}^{D}X_{i}^{C\cap U}X_{i}^{C}$ be a vector space basis of $C_{i}$. Then $X_{i}^{C}X_{i}^{C\cap U}X_{i}^{D}X_{i}^{B}$ is a vectorspace basis of $\langle CB\rangle$.

Let $X_{i}^{D}X_{i}^{C\cap U}X_{i}^{B}X_{i}^{U}$ be a vector space basis of $U_{i}$. Let $X_{i}^{D}X_{i}^{C\cap U}X_{i}^{B}X_{i}^{U}X_{i}^{C}X_{i}^{V}$ be a vector space basis of $V_{i}$.

To get a vector space basis of $\langle C\bar{a}_{1}\rangle_{2}$ over $C_{2}$, we choose $Y_{2}^{D}Y_{2}^{C\cap U}Y_{2}^{C}$ linearly independent, such that $Y_{2}^{D}$ is a vector space basis of $\langle D\bar{a}_{1}\rangle_{2}$ over $C_{2}$, $Y_{2}^{C\cap U}$ is a vectorspace basis of $\langle(C\cap U)\bar{a}_{1}\rangle_{2}$ over $C_{2}Y_{2}^{D}$, and $Y_{2}^{C}$ is a vector space basis of $\langle C\bar{a}_{1}\rangle_{2}$ over $C_{2}Y_{2}^{D}Y_{2}^{C\cap U}$. By (1) $Y_{2}^{D}Y_{2}^{C\cap U}Y_{2}^{C}$ is a vector space basis over $V_{2}$.

By Corollary 3.9 and (1) we get that the union of the following sets is a vector space basis for $\langle V\bar{a}\rangle_{2}$:

$X_{2}^{D}X_{2}^{C\cap U}X_{2}^{C}X_{2}^{B}X_{2}^{U}X_{2}^{V}$ for $V_{2}$,

$X_{2}^{D}X_{2}^{C\cap U}X_{2}^{C}Y_{2}^{D}Y_{2}^{C\cap U}Y_{2}^{C}\bar{e}_{2}$ for $\langle C\bar{a}\rangle_{2}$, and

$\{[x,y]:x\in X_{1}^{B}X_{1}^{U}X_{1}^{V},y\in\bar{a}_{1}\}$.

If we use (2) and Corollary 3.9,

then we get the following vector space basis for $\langle U\bar{a}\rangle_{2}$, as the union of:

$X_{2}^{D}X_{2}^{B}X_{2}^{C\cap U}X_{2}^{U}$ for $U_{2}$,

$X_{2}^{D}X_{2}^{B}Y_{2}^{D}\{[x,y]:x\in X_{1}^{B},y\in\bar{a}_{1}\}\bar{e}_{2}\bar{e^{\prime}}_{2}$ for $\langle B\bar{a}\rangle_{2}$, and

$\{[x,y]:x\in X_{1}^{U}X_{1}^{U\cap C},y\in\bar{a}_{1}\}$.

We can rewrite the vector space basis as the union of:

$X_{2}^{D}X_{2}^{C\cap U}X_{2}^{B}X_{2}^{U}$ for $U_{2}$,

$X_{2}^{D}Y_{2}^{D}\bar{e}_{2}\bar{e^{\prime}}_{2}$ for $\langle D\bar{a}\rangle_{2}$, and

$\{[x,y]:x\in X_{1}^{B}X_{1}^{U}X_{1}^{U\cap C},y\in\bar{a}_{1}\}$.

By Corollary 3.9 we have $\langle U^{*}\bar{a}^{*}\rangle=U^{*}\otimes_{D^{*}}\langle D^{*}\bar{a}^{*}\rangle$.

Note, that $Y_{2}^{C\cap U}=\{[x,y]:x\in X_{1}^{C\cap U},y\in\bar{a}_{1}\}$.

Furthermore we have shown, that $Y_{2}^{B}=\{[x,y]:x\in X_{1}^{B},y\in\bar{a}_{1}\}$ is a vector space basis of $\langle B\bar{a}_{1}\rangle_{2}$ over $\langle B_{2}Y_{2}^{D}\rangle_{2}$.

We consider $e\in\bar{e^{\prime}}_{3}\bar{e^{\prime\prime}}_{3}$. Then w.l.o.g.

[TABLE]

where $v_{3}\in V_{3}$ and $\Delta$ is a linear combination of monomials of the form $[w,z]$ with $w\in V_{2},z\in\bar{a}_{1}$ or $z\in\bar{a}_{2},w\in V_{1}$ or $[[v,y],u]$ where $v,u\in V_{1},y\in\bar{a}_{1}$ or $[[x,v],z]$ where $x,z\in\bar{a}_{1},v\in V_{1}$.

By Corollary 3.9 we get that $v,u,w$ and $v_{3}$ are in $C$ similarly as above.

If $e\in\bar{e^{\prime\prime}}_{3}$, then (1) and (2) imply, that all these $u,v,w$ and $v_{3}$ are in $D$.

Hence we can replace (1) by $\langle V\bar{a}\rangle=V\otimes_{C}\langle C\bar{e}\rangle$ and (2) by $\langle U\bar{a}\rangle=U\otimes_{B}\langle B\bar{e}\bar{e^{\prime}}\rangle$.

Now we define analogously as above, that $Y_{3}^{D}$ is a vector space basis of $\langle D\bar{a}_{1}\bar{e}_{2}\rangle_{3}$ over $C_{3}$ , $Y_{3}^{C\cap U}$ is a vector space basis of $(\langle C\cap U)\bar{a}_{1}\bar{e}_{2}\rangle_{3}$ over $C_{3}Y_{3}^{D}$, and $Y_{3}^{C}$ is a vectorspace basis of $\langle C\bar{a}_{1}\bar{e}_{2}\rangle$ over $C_{3}Y_{3}^{D}Y_{3}^{C\cap U}$. Hence $Y_{3}^{D}Y_{3}^{C\cap U}Y_{3}^{C}$ is a vector space basis of $\langle C\bar{a}_{1}\bar{e}_{2}\rangle_{3}$ over $C_{3}$ and by (1) over $V_{3}$. We assume, that $\bar{a}_{1}$ is ordered by $a_{1}^{j}<a_{1}^{k}$ for $j<k$. Furthermore we order $X_{1}^{B}X_{1}^{U}X_{1}^{V}$.

By (1) and Corollary 3.9 the union of the following sets is a vector space basis (5) of $\langle V\bar{a}\rangle_{3}$:

$X_{3}^{D}X_{3}^{C\cap U}X_{3}^{C}X_{3}^{B}X_{3}^{U}X_{3}^{V}$ for $V_{3}$,

$X_{3}^{D}X_{3}^{C\cap U}X_{3}^{C}Y_{3}^{D}Y_{3}^{C\cap U}Y_{3}^{C}\bar{e}_{3}$ for $\langle C\bar{a}\rangle_{3}$,

$\{[x,y]:x\in X_{2}^{B}X_{2}^{U}X_{2}^{V},y\in\bar{a}_{1}\;or\;x\in Y_{2}^{D}Y_{2}^{C\cap U}Y_{2}^{C}\bar{e}_{2},y\in X_{1}^{B}X_{1}^{U}X_{1}^{V}\}$,

$\{[[a_{1}^{j},z],a_{1}^{k}]:j\leq k,z\in X_{1}^{B}X_{1}^{U}X_{1}^{V}\},$

$\{[[x,a_{1}^{j}],y]:x\leq y\in X_{1}^{B}X_{1}^{U}X_{1}^{V}\}$.

Let $Y_{3}^{B}$ be a vector space basis of $\langle B\bar{a}\rangle_{3}$ over $X_{3}^{D}X_{3}^{B}Y_{3}^{D}$. By (2) und (5) we have

[TABLE]

[TABLE]

We apply Corollary 3.9 to (2) to get the union of the following sets as a vector space basis (6) of $\langle U\bar{a}\rangle_{3}$:

$X_{3}^{D}X_{3}^{B}X_{3}^{C\cap U}X_{3}^{U}$ for $U_{3}$,

$X_{3}^{D}X_{3}^{B}Y_{3}^{D}Y_{3}^{B}$ for $\langle B\bar{a}\rangle_{3}$, and

$\{[x,y]:x\in X_{2}^{C\cap U}X_{2}^{U},y\in\bar{a}_{1}\;or\;x\in Y_{2}^{D}Y_{2}^{B}\bar{e}_{2}\bar{e^{\prime}}_{2},y\in X_{1}^{C\cap U}X_{1}^{U}\}$

$\{[[a_{1}^{j},z],a_{1}^{k}]:j\leq k,z\in X_{1}^{C\cap U}X_{1}^{U}\},$

$\{[[x,a_{1}^{j}],y]:x\leq y\in X_{1}^{C\cap U}X_{1}^{U}\}$.

We can rewrite this vector space basis using the structure of $Y_{2}^{B}$ and $Y_{3}^{B}$, as described above:

$X_{3}^{D}X_{3}^{B}X_{3}^{C\cap U}X_{3}^{U}$ for $U_{3}$,

$X_{3}^{D}Y_{3}^{D}\bar{e}_{3}\bar{e^{\prime}}_{3}$ for $\langle D\bar{a}\rangle_{3}$, and

$\{[x,y]:x\in Y_{2}^{D}\bar{e}_{2}\bar{e^{\prime}}_{2},y\in X_{1}^{B}X_{1}^{C\cap U}X_{1}^{U}\;or\;x\in X_{2}^{B}X_{2}^{C\cap U}X_{2}^{U},y\in\bar{a}_{1}\}\\ \{[[a_{1}^{j},z],a_{1}^{k}]:j\leq k,z\in X_{1}^{B}X_{1}^{C\cap U}X_{1}^{U}\}\\ \{[[x,a_{1}^{j}],z]:x\leq z\in X_{1}^{B}X_{1}^{C\cap U}X_{1}^{U}\}$.

By Corollary 3.9 we get

[TABLE]

∎

We consider further consequences of Corollary 3.9.

Corollary 3.11**.**

*Let $B$ and $\langle x\rangle$ with $U_{i}(x)$ be finite graded 3-nilpotent Lie algebras. Let $Z=Z_{1}Z_{2}Z_{3}$ be an ordered vectorspace basis of $B$.

If $H$ is the ideal generated by $x$ in $D=B\otimes\langle x\rangle$, then it is freely generated by the following set $X$:*

1. (1)

If $U_{3}(x)$,then $X=X_{3}=\{x\}$. 2. (2)

If $U_{2}(x)$, then $X=\{x\}\cup\{[z,x]:z\in Z_{1}\}$. 3. (3)

If $U_{1}(x)$, then $X=\{x\}\cup\{[z,x]:z\in Z_{2}\}\cup\{[[z_{1},x],z_{2}]:z_{1}\leq z_{2}\}$.

The underlying vectorspace of $D=B\otimes\langle x\rangle$ is $B\oplus H$. In case (1) and (2) $X$ is also a vectorspace basis of $H$ and in case (3) $X\cup\{[[x,z],x]:z\in Z_{1}\}$ is a vector space basis of $H$.

The algebraic background is well known. See e.g. [2].

Lemma 3.12**.**

Assume $B\in\mathcal{K}_{3}$, $\langle x\rangle\in\mathcal{K}_{3}$ with $U_{i}(x)$, and $D=B\otimes\langle x\rangle$. Then

1. (1)

$B\leq D$, $\delta(D)=\delta(B)+1$. 2. (2)

$D\in\mathcal{K}_{3}$.

Proof.

As above let $H$ be the ideal generated by $x$ in $D=B\otimes\langle x\rangle$.

ad1) Let $B\subseteq E\subseteq D$. Let $Y$ be a generating o-system for $B$. Then $B\simeq F(Y)/I$ and $D\cong F(Yx)/\langle\langle I\rangle\rangle^{F(Yx)}$. For $E$ we choose $W$ in $H$ such that $YW$ is a generating o-system for $E$. Then $E\cong F(YW)/\langle\langle I\rangle\rangle^{F(YW)}$. Hence $\delta(B)\leq\delta(E)$, $\delta_{2}(B)\leq\delta_{2}(E)$, and $\delta(D)=\delta(B)+1$.

ad2) Now we consider any $E=\langle E_{1}E_{2}\rangle\subseteq D$. There is an o-system $W$ over $B\cap E$, that generates $E$ over $E\cap B$. Then $W$ extends every o-system for $B\cap E$ to a generating o-system for $E$. We have $\delta(E)=\delta(B\cap E)+\mid W\mid$ .

∎

Definition 3.13**.**

Assume $b,e\in B$ with $U_{i}(b)$, $U_{j}(e)$, and $i<j\leq c$ and there is no solution of $[b,x]=e$ in B. Such a pair $b,e$ is called a divisor problem for $B$.

Definition 3.14**.**

$B(e:b)$ is $B\otimes\langle x\rangle$ factorized by $[b,x]=e$.

We describe another way to define $B(e:b)$ for $B\in\mathcal{K}_{c}$. Let $\langle bex\rangle$ be the graded Lie algebra defined by $U_{j-i}(x)$, and $[b,x]=e$. Then $\langle bex\rangle\in\mathcal{K}_{c}$, $\langle be\rangle\leq\langle bex\rangle$, and $\delta(\langle be\rangle)=\delta(\langle bex\rangle)$. By Theorem 3.4 $B\otimes_{\langle be\rangle}\langle bex\rangle$ exists. It is $B(e:b)$.

Now we consider again $\mathcal{K}_{3}$. As above we assume that $Z$ is an ordered vector space basis of $B$. Let $b$ be the first element of $Z$ and $e$ the second. We use the description of $B\otimes\langle x\rangle$ above. Let $X^{-}$ be the subset of all elements of $X$, where $b$ does not occure. Let $H^{-}$ be the graded Lie algebra freely generated by $X^{-}$.

Note that the underlying vectorspace of $B(e:b)$ is $B\oplus H^{-}$ and the Lie multiplication is given the Lie multiplication in $B$ and in $H^{-}$ and by the action of $B$ on $H^{-}$ in $B\otimes\langle x\rangle$ factorized by $[b,x]=e$.

Lemma 3.15**.**

Assume $B\in\mathcal{K}_{3}$, $b\in U_{i}(B)$, $e\in U_{j}(B)$, $1\leq i<j\leq 3$, and there is no solution of $[b,?]=e$ in $B$.

1. (1)

$B(e:b)\cong B\otimes_{\langle b,e\rangle}\langle b,e,x\rangle$, definded as above. 2. (2)

The underlying vector space of $B(e:b)$ is $B\oplus H^{-}$. If $j-i=2$, then $X^{-}$ is also a vector space basis of $H^{-}$. If $j-i=1$, then $X^{-}\cup\{[[x,z],x]:z\in Z_{1}\setminus\{b\}\}$ is a vectorspace basis of $H^{-}$. 3. (3)

If $E\subseteq B(e:b)$, then $\delta_{i}(E)\geq\delta_{i}(E\cap B)$ and $\delta(E)\geq min\{2,o-dim(E)\}$. 4. (4)

$B\leq B(e:b)$, $\delta(B(e:b)=\delta(B)$. 5. (5)

$B(e:b)$* is in $\mathcal{K}_{3}$.*

Proof.

ad (1) For every suitable $A=\langle Ba\rangle$ with $[b,a]=e$ there is a homomorphism $h$ of $B\otimes_{\langle be\rangle}\langle bex\rangle$ onto $A$ with $h(B)=B$ pointwise and $h(x)=a$.

ad (2) is clear.

ad (3) $D=B\otimes\langle x\rangle$. $Z$, $H$ , $X$ , $X^{-}$,and $H^{-}$ are defined as above. By definition $B(e:b)=D/\langle\langle[x,b]-e\rangle\rangle$. Let $E$ be a subspace of $B(e:b)$.

If $b\notin E$, then there is a subset $W$, such that for every o-sytem $V$ of $B\cap E$ the set $V\cup W$ is a generating o-system for $E$ and $\delta(E)=\delta(E\cap B)+\mid W\mid$. For $\delta_{2}$ use a similar argument. The assertion is proved for this case.

Now we assume that $b\in E$. First we assume that there is an element $d\in U_{j-i}(E)\setminus B$. Then w.l.o.g. $d=x+c$ with $c\in U_{j-i}(B)$. Then $E=\langle(E\cap B)dW\rangle$, where $VW$ is an o-system for $E$, if $V$ is an o-system for $\langle(E\cap B)d\rangle$. We get $\delta(B\cap E)+\mid W\mid)\leq\delta(E)$.

If $o-dim(B\cap E)=1$, then $B\cap E=\langle b\rangle$. Then $\delta(E)=2+\mid W\mid$.

Finally $b\in E$ and $U_{j-i}(E)=U_{j-i}(E\cap B)$. By similar considerations as above $\delta(E)=\delta(E\cap B)+\mid W\mid$. Similar considerations work for $\delta_{2}$.

(4) and (5) follow from (3).

∎

Lemma 3.16**.**

If $B\in\mathcal{K}_{2}$, then $B(e:b)$ is in $\mathcal{K}_{2}$.

Lemma 3.17**.**

Assume $A\cap C=B$ are substructures in a 3-nilpotent graded Lie algebra $M$ over a field. Furthermore we define for $c\in C$ that $A^{+}=\langle Ac\rangle=A\otimes_{B}\langle Bc\rangle$ and $B^{+}=\langle Bc\rangle^{C}$. Then the following are equivalent:

1. (1)

$\langle AC\rangle=A\otimes_{B}C$. 2. (2)

$\langle AC\rangle=\langle A^{+}C\rangle=A^{+}\otimes_{B^{+}}C$.

Proof.

Assume (1): A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{B}C. By Mon we have A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{\langle Bc\rangle}C. This is $\langle AC\rangle=\langle A^{+}C\rangle=A^{+}\otimes_{B^{+}}C$.

For the other direction we have A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{\langle Bc\rangle}C and A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{B}\langle c\rangle. By **Trans ** we get A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{B}C, as desired. ∎

We will apply the Lemma in $\mathcal{K}_{3}$, where $[b,c]=e$ for homogeneous $b,e$ in $B$.

4. 2-nilpotent graded Lie algebras

We work in $\mathcal{K}_{2}$. $\delta$ is $\delta_{2}$. In Lemma 2.8 submodularity for $\delta$ is shown. Then Lemma 2.11 implies:

Lemma 4.1**.**

Let $M$ be in $\mathcal{K}_{2}$. Then for all substructures $A,C$ finite and $E$ of $M$ the following is true:

1. (1)

If $C\leq M$ and $E\subseteq M$, then $E\cap C\leq E$. 2. (2)

If $A\leq C\leq M$, then $A\leq M$. 3. (3)

If $A,C\leq M$, then $A\cap C\leq M$. 4. (4)

If $A\leq^{k}C\leq M$, then $A\leq^{k}M$.

Proof.

For (4) we need some proof. Assume $A\subseteq E\subseteq M$ and $o-dim_{1}(E/A)\leq k$ or equivalently $ldim(E_{1}/A_{1})\leq k$. By (1) $\delta(E\cap C)\leq\delta(E)$. Since $o-dim_{1}(E\cap C)/A)\leq k$, we have $\delta(A)\leq\delta(E\cap C)$.

∎

The lemma implies the existence of $CSS_{2}(A)$ in $M$ - the smallest strong subspace that contains $A$.

Lemma 3.16 implies that for every $A\in\mathcal{K}_{2}$ there is some $A\subseteq C\in\mathcal{H}_{2}$, such that $A\leq C$, $\delta(A)=\delta(C)$, and $C$ is obtained by free adjunction of divisors.

All results for $c=2$ in this paper are proved in [4], exept Lemmas 4.7 and 4.8. In [6] different proofs are given, using a general method developed in that paper, to obtain the additive collaps. In both papers we work in $\mathcal{H}_{2}$. Here we use $\mathcal{K}_{2}$. For $c=2$ there is no big difference between the two approches. But for greater $c$ we have to start with $\mathcal{K}_{c}$.

In this section we summarize some results from [4]. By Definition 2.15 $B\leq^{ldim(A/B)+n}C$ means that $\delta(B)\leq\delta(E)$ for all $B\subseteq E\subseteq C$ with $lin-dim(E_{1}/B_{1}\leq ldim(A/B)+n$.

Theorem 4.2**.**

1. (1)

$\mathcal{K}_{2}$* has the amalgamation property for strong embeddings.* 2. (2)

If $A,B,C\in\mathcal{K}_{2}$, $B\leq A$,and $B\leq^{ldim(A/B)+n}C$, then there is an amalgam $D$ of $A$ and $C$ over $B$ in $\mathcal{K}_{2}$, such that $C\leq D$ and $A\leq^{n}D$. If no divisor problem in $B$ has solutions in both $A$ and $C$, then the free amalgam $D=A\otimes_{B}C$ fulfils the assertion. In this case $\delta_{2}(D)=\delta_{2}(A)+\delta_{2}(C)-\delta_{2}(B).$

Using this amalgamation we obtain the Fraïssé Hrushovski Limit $\mathfrak{M}$ in the next theorem.

Theorem 4.3**.**

There is a countable structure $\mathfrak{M}$ in $\mathcal{K}_{2}$, that satisfies the following condition:

**rich: **

If $B\leq A$ in $\mathcal{K}_{2}$ and there is a strong embedding $f$ of $B$ in $\mathfrak{M}$, then there is a strong extension of $f$ that maps $A$ into $\mathfrak{M}$.

$\mathfrak{M}$* is uniquely determined up to isomorphisms.*

If $b\in\mathfrak{M}_{1}$ and $e\in\mathfrak{M}_{2}$, then there exists $B\leq\mathfrak{M}$ with $b,e\in B$. Hence there is a strong embedding of $B(e:b)$ over $B$ into $\mathfrak{M}$. That means $\mathfrak{M}$ is closed under homogeneous divisors and $\mathfrak{M}\in\mathcal{H}_{2}$. We speak about rich structures.

Theorem 4.4**.**

If $M$ and $N$ are rich $\mathcal{K}_{2}$-structures, $\langle\bar{a}\rangle\leq M$, $\langle\bar{b}\rangle\leq N$, and $\langle\bar{a}\rangle\cong\langle\bar{b}\rangle$, then

[TABLE]

By the Theorem above there is a complete elementary theory $T_{2}$ of all rich $\mathcal{K}_{2}$-structures. $T_{2}=Th(\mathfrak{M})$. Let $T_{2}(1)$ be an elementary description of $\mathcal{K}_{2}$ and

$T_{2}(2)$**: **

For all n and all $B\leq A$ in $\mathcal{K}_{2}$ there is an elementary sentences saying that every restricted by $(n+ldim(A/B))$ strong embedding $f$ of $B$ in $M$ can be extended to an restricted by n strong embedding of $A$ in $M$.

The next theorem imlies that $T_{2}(1)\cup T_{2}(2)$ is an elementary axiomatization of $T_{2}$.

Theorem 4.5**.**

1. (1)

A rich $\mathcal{K}_{2}$-structure satisfies $T_{2}(1)\cup T_{2}(2)$. 2. (2)

Let $M$ be a model of $T_{2}(1)$. $M$ is rich if and only if M is an $\omega$ - saturated model of $T_{2}(2)$.

Later we need the following:

Definition 4.6**.**

We work in some $M\in\mathcal{K}_{2}$. Let $A$, $C$, and $N$ be substrucures, where $A$ and $C$ are finite.

1. (1)

$\delta_{2}(A/C)=\delta_{2}(\langle AC\rangle)-\delta_{2}(C)$. 2. (2)

$\delta_{2}(A/N)=min\{\delta_{2}(A/C):C\subseteq N,Cfinite,\langle AC\rangle\cap N=C\}.$

Note that in (1) $\delta_{2}(A/C)$ is not equal to

$o-dim(A/C)-ideal-dim(A/C)$ in general.

In general we have:

$o-dim(A/N)\leq o-dim(A)$.

For $B\subseteq A$ it holds $o-dim(A)-o-dim(B)\leq o-dim(A/B)$.

Lemma 4.7**.**

*Let $M$ be a 2-nilpotent graded Lie algebra in $\mathcal{K}_{2}$. There exists a function $h(n)$ such that:

We consider substructures $C$, $A$, and $B=C\cap A$ with $o-dim(A/B)\leq n$ and $\delta_{2}(A/C)\geq 0$. Then there exist $D$, $K$, $D\cap K=H$, and $X_{C}\subseteq C_{2}\cap\langle C_{1}A_{1}\rangle$ linearly independent over $\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2}+B_{2}$, such that*

1. (1)

$B\subseteq H\subseteq D\subseteq C$, $D_{1}=C_{1}$, $C=\langle DX_{C}\rangle$, $K=\langle AH\rangle$, and $o-dim(H/B)\leq h(n)$. 2. (2)

$\langle CA\rangle=\langle DA\rangle=D\otimes_{H}\langle HA\rangle$. 3. (3)

We can define $h(n)=(n+1)n$.

Proof.

We define:

$r_{A}=ldim((\langle A_{1}C_{1}\rangle_{2}\cap A_{2})/\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$,

$r_{C}=ldim((\langle A_{1}C_{1}\rangle_{2}\cap C_{2})/\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$,

$r_{B}=ldim((\langle A_{1}C_{1}\rangle_{2}\cap B_{2})/\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$, and

if $\langle A_{1}C_{1}\rangle=F(A_{1}C_{1})/I$, then $s=ldim(I/(I\cap(F(A_{1})+F(C_{1}))$.

By Corollary 2.9 we get

$\delta_{2}(\langle AC\rangle)-\delta_{2}(C)=\delta_{2}(A)-\delta_{2}(B)-(r_{A}+r_{C}-r_{B}+s).$

Since $\delta_{2}(A/C)\geq 0$, we have $\delta_{2}(A/B)\geq 0$ and it is a bound for the size of $r_{A}+r_{C}-r_{B}+s$.

We consider an ideal basis $\{\psi_{1},\ldots,\psi_{s}\}$ of $I$ over $I\cap(F(A_{1})+F(C_{1}))$. Let $\{a_{i}^{1}\in A_{1}:1\leq i\leq k\}$ be a vector basis for $\langle CA\rangle_{1}$ over $C_{1}$.

$\psi_{i}$ has the following form:

[TABLE]

where $b_{l}\in C_{1}$ and $c_{i}\in F(C_{1})_{2}$. We add all the $b_{l}$ and $c_{i}$ to $B$ and obtain $H^{0}$.

Let $d_{1},\ldots,d_{r}$ with $r=r_{A}+r_{C}-r_{B}$ be elements in $\langle A_{1}C_{1}\rangle_{2}$ linearly independent over $\langle A_{1}\rangle_{2}+\langle C_{1}\rangle_{2}$, such that

$d_{1};\ldots,d_{r_{B}}$ are in $B_{2}$ , $d_{r_{B}+1},\ldots,d_{r_{A}}$ are in $A_{2}$ linearly independent over $B_{2}$ and $d_{r_{A}+1},\ldots,d_{r}$ are in $C_{2}$ linearly independnet over $B_{2}$ . For $1\leq i\leq r_{A}$

[TABLE]

where $c^{1}_{l}\in C_{1}$, $a^{1}_{l}\in A_{1}$, $a^{2}_{i}\in\langle A_{1}\rangle$, and $c^{2}_{i}\in\langle C_{1}\rangle$. We add all these $c^{1}_{l}$ and $c^{2}_{i}$ to $H^{0}$ and obtain $H$. Define $\{d_{r_{A}+1},\ldots,d_{r}\}=X_{C}$. We choose $Y_{C}\subseteq C_{2}$ maximal linearly independent over $\langle C_{1}A_{1}\rangle_{2}$ and define $D=\langle C_{1}HY_{C}\rangle$. By Corollary 3.8 it follows the assertion:

[TABLE]

We have added $(k+1)\times(s+r_{A})$ many elements to $B$. $k+1\leq n+1$ and $s+r_{A}\leq\delta_{2}(A)-\delta_{2}(B)\leq o-dim(A)-o-dim(B)\leq o-dim(A/B)\leq n$.

∎

Lemma 4.8**.**

*Let $M$ be a 2-nilpotent graded Lie algebra in $\mathcal{K}_{2}$.

We consider finite substructures $C$, $A$, and $B=C\cap A$ with $o-dim_{1}(A/B)=n$ and $C\leq M$. Then there exists $D$, $H$, $K$, and $X_{C}X_{B}X_{A}$, such that*

1. (1)

$H\subseteq D\subseteq C$*, $B_{1}\subseteq H$, $D_{1}=C_{1}$, $D\cap K=H$ and *

$o-dim(H/B)\leq h(n)=(n+1)\times n$, 2. (2)

$H\subseteq K\subseteq\langle HA\rangle$, $K_{1}=\langle HA\rangle_{1}$, 3. (3)

$\langle CA\rangle=\langle DK\rangle=D\otimes_{H}\langle K\rangle$, 4. (4)

$X_{C}X_{B}X_{A}\subseteq\langle C_{1}A_{1}\rangle_{2}$* is linearly independent over $\langle D_{1}\rangle_{2}+\langle K_{1}\rangle_{2}$, $X_{B}\subseteq B_{2}$, $\langle HX_{B}\rangle=\langle HB\rangle$,* 5. (5)

$X_{C}\subseteq C_{2}$, $\langle DX_{B}X_{C}\rangle=C$, and 6. (6)

$X_{A}\subseteq A_{2}$, $\langle KX_{B}X_{A}\rangle=\langle HA\rangle$.

Proof.

We use the following subalgebras of $\langle CA\rangle$:

$B^{0}=\langle B_{1},(B_{2}\cap(\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2})),Y_{B}\rangle$,where $Y_{B}$ is a maximal subset of $B_{2}$ linearly indepentent over $\langle C_{1}A_{1}\rangle_{2}$.

$C^{0}=\langle C_{1},B^{0},Y_{B},Y_{C}\rangle$, where $Y_{C}$ is a maximal subset of $C_{2}$ linearly independent over $\langle C_{1}A_{1}Y_{B}\rangle_{2}$.

$A^{0}=\langle A_{1},B^{0},Y_{B},Y_{A}\rangle$, where $Y_{A}$ is a maximal subset of $A_{2}$ linearly independent over $\langle C_{1}A_{1}Y_{B}\rangle_{2}$. Furthermore

$X_{B}$ is a maximal subset of $B_{2}\cap\langle C_{1}A_{1}\rangle_{2}$ linearly independent over $B_{2}\cap(\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2})$.

$X_{C}$ is a maximal subset of $C_{2}\cap\langle C_{1}A_{1}\rangle_{2}$ linearly independent over $C_{2}\cap(\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2}+X_{B})$.

$X_{A}$ is a maximal subset of $C_{A}\cap\langle C_{1}A_{1}\rangle_{2}$ linearly independent over $A_{2}\cap(\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2}+X_{B})$. Then

$B=\langle B^{0}X_{B}\rangle$, $C=\langle C^{0}X_{B}X_{C}\rangle$, and $A=\langle A^{0}X_{B}X_{A}\rangle$.

Since $\langle CA\rangle=\langle C^{0}A^{0}\rangle$ and $\delta_{2}(C)=\delta_{2}(C^{0})+\mid X_{B}X_{C}\mid$, we have $0\leq\delta_{2}(A/C)\leq\delta_{2}(A^{0}/C^{0})$. Furthermore $C^{0}\cap A^{0}=B^{0}$. Then we get as in the proof of Lemma 4.7

[TABLE]

where $s$ is obtain as above:

If $\langle A_{1}C_{1}\rangle=F(A_{1}C_{1})/I$, then $s=ldim(I/(I\cap(F(A_{1})+F(C_{1}))$. Since $C\leq M$, we have $\langle C_{1}\rangle\leq M$. Then

[TABLE]

Hence $s\leq lin-dim(A_{1}/B_{1})$.

As above we add at most $(lin-dim(A_{1}/B_{1})+1)\times s$ many elements from $C^{0}$ to $B^{0}$ to obtain $H^{0}$ such that

[TABLE]

We define $H=H^{0}$, $D=\langle C^{0}H\rangle$, and $K=\langle HA^{0}\rangle$ and obtain $\langle DK\rangle=D\otimes_{H}K$.

By construction $o-dim(H/B)\leq(n+1)n$.

∎

5. The functor $\mathbb{F}$

If $A$ is in $\mathcal{K}_{c}$, then $A^{*}=A/A_{c}$ is in $\mathcal{K}_{c-1}$ and the canonical homomorphism $*$ of $A$ onto $A^{*}$ is injectiv on $A_{1}\oplus\ldots\oplus A_{c-1}$ Let $\tau$ be this injectiv map of $A_{1}\oplus\ldots\oplus A_{c-1}$ onto $A^{*}$. Let $F(X)$ be the free graded c-nilpotent Lie algebra over $K$ freely generated by an o-system $X$ , where $X_{c}=\emptyset$. Then $F(X)^{*}$ is the free graded (c-1)-nilpotent Lie algebra over $K$ freely generated by $\tau(X)$.

Now we assume, that $A\in\mathcal{K}_{c-1}$, $F(X)\in\mathcal{K}_{c}$ where $\tau(X)$ corresponds to a generating o-system of $A$ and $X_{c}=\emptyset$. Then $A=F(X)^{*}/I$ for some $I$ in $F(X)^{*}$. That means $F(X)^{*},I$ is the canonical pair for $A$ in $\mathcal{K}_{c-1}$. Let $J$ be the ideal in $F(X)$ generated by $\tau^{-1}(I)$. We define:

Definition 5.1**.**

$\mathbb{F}(A)=F(X)/J$.

Let $B\in\mathcal{K}_{c}$ and $B=\langle B_{1},\ldots,B_{c-1}\rangle$. If $A\in\mathcal{K}_{c-1}$ and $A=B^{*}$, then there is a homomorphism of $\mathbb{F}(A)$ onto $B$. In the case $A\in\mathcal{K}_{2}$ $A^{*}$ is a vector space and $\mathbb{F}(A^{*})$ is the free 2-nilpotent Lie algebra over $A^{*}$.

$\mathbb{F}$ is a functor from $\mathcal{K}_{c-1}$ with embeddings into $\mathcal{K}_{c}$ with homomorphisms. Let $B$ and $A$ be in $\mathcal{K}_{c-1}$ and $f:B\rightarrow A$ be an embedding. W.l.o.g. $B$ is a substructure of $A$ :

We consider the canonical pair $F(X)^{*},I$ for $A$ as above. By Lemma2.13 there exits a subalgebra $H$ of $F(X)^{*}$ such that $H,I\cap H$ is the canonical pair for $B$. Let $K$ be the subalgebra of $F(X)$ generated by $\tau^{-1}(H)$. Then $\mathbb{F}(B)=K/J_{B}$ and $\mathbb{F}(A)=F(X)/J$ where $J_{B}=\langle\langle\tau^{-1}(I\cap H)\rangle\rangle^{K}$ and $J$ as above. Then $(J_{B})_{i}=J_{i}\cap K_{i}$ for $i<c$ and $(J_{B})_{c}\subseteq(J_{c}\cap K_{c})$. This gives the desired homomorhism $\mathbb{F}(f)$ of $\mathbb{F}(B)$ into $\mathbb{F}(A)$. For $c=2$ this is an embedding.

Already for $c=3$ there are examples where $\mathbb{F}(f)$ is not an embedding. See [1]. This causes a lot of efforts. We use the notation $\gamma_{B}^{A}$ for $\mathbb{F}(f)$.

Consicer $A\in\mathcal{K}_{c}$. Let $A^{-}$ be $\langle A_{1}\oplus\ldots\oplus A_{c-1}\rangle^{A}$. Then there is some $N(A)\subseteq\mathbb{F}(A^{*})_{c}$, such that $A^{-}=\mathbb{F}(A^{*})/N(A)$. Hence

[TABLE]

Furthermore $M\in\mathcal{K}_{c}$ implies $\mathbb{F}(M^{*})\in\mathcal{K}_{c}$: This is a consequence of the next inequality. We consider $A_{i}$ as subspace in $M_{i}$ and in $\mathbb{F}(M^{*})_{i}$. Then

[TABLE]

For $\delta_{2}$ submodularity is true (Lemma 2.8). For $A\in\mathcal{K}_{c}^{fin}$ and $A\neq\langle 0\rangle$ we have $\delta_{2}(A)>0$. Assume $A\subseteq M\in\mathcal{K}_{c}$. By Corollary2.12 $\bigcap\{C:A\subseteq C\leq_{2}M\}$ exists. It is the smallest 2-strong substructure of $M$, that contains $A$. We call it $CSS_{2}^{M}(A)$ or short $CSS_{2}(A)$.

Lemma 5.2**.**

Assume $D\leq_{2}M\in\mathcal{K}_{c}$ and $XY$ is a vector space basis of $D_{2}$ over $\langle D_{1}\rangle$. Then $\langle D_{1}X\rangle\leq_{2}M$. Especially $\langle D_{1}\rangle\leq_{2}M$.

Proof.

Assume $\langle D_{1}X\rangle\subseteq E$. There is $Y_{0}\subseteq Y$ linearly independent over $E_{2}$, such that $D_{2}=\langle(D_{2}\cap E_{2})Y_{0}\rangle^{lin}$. Then

[TABLE]

and

[TABLE]

Hence

[TABLE]

∎

In the next lemma we describe the structure of $CSS_{2}(A)$ for $A\in\mathcal{K}_{2}$:

Lemma 5.3**.**

Let $B$ be a substructure of $M\in\mathcal{K}_{2}$. Then $CSS_{2}(B)=D=\langle D_{1}B_{2}\rangle$ and the following is true:

1. (1)

$B_{1}\subseteq D_{1}$, $B_{2}\subseteq D_{2}$ and $\langle D_{1}\rangle\leq_{2}M$. 2. (2)

$CSS_{2}(\langle B_{1}\rangle)=C=\langle C_{1}\rangle\leq_{2}M$* and $C_{1}\subseteq D_{1}.$* 3. (3)

If $C_{1}\not=D_{1}$, then

[TABLE]

Proof.

(1) is true by Lemma 5.2 and the definitions.

The first statement of (2) follows again from Lemma 5.2. If $C_{1}\not\subseteq D_{1}$, then

[TABLE]

by Lemma 2.8. Then

[TABLE]

since $B_{1}\subseteq C_{1}\cap D_{1}$, a contradiction.

If the inequality in (3) is not true, then $D=\langle C_{1}B_{2}\rangle$.

∎

Lemma 5.4**.**

Let $A$ be a 2-nilpotent graded Lie algebra with $A\in\mathcal{K}_{2}$. Assume that $X\subseteq A_{1}$ is a vector space basis of $A_{1}$ and $Y\subseteq A_{2}$ is a vector space basis of $A_{2}$ over $\langle X\rangle_{2}$. We consider $X$ and $Y$ also as subsets in $\mathbb{F}(A)$. Then $\langle XY\rangle^{\mathbb{F}(A)}_{3}$ is freely generated over $\langle X\rangle^{\mathbb{F}(A)}_{3}$ by $\{[x,y]:x\in X,y\in Y\}$. In other words $\mathbb{F}(A)=\langle XY\rangle^{\mathbb{F}(A)}=\langle X\rangle^{\mathbb{F}(A)}\otimes\langle Y\rangle^{\mathbb{F}(A)}$.

In this paper we mainly consider the case $c=3$.

The next Theorem gives an upper bound for the size of the kernel of $\gamma_{B}^{A}$ for $A,B$ in $\mathcal{K}_{2}^{fin}$. It is essentially in [1].

Theorem 5.5**.**

Let $B\subseteq A$ be 2-nilpotent graded Lie algebras in $\mathcal{K}_{2}^{fin}$, where $\mathbb{F}(A)\in\mathcal{K}_{3}$. Then

**a): **

If $B\leq_{2}A$, then $\gamma_{B}^{A}$ is an embedding of $\mathbb{F}(B)$ into $\mathbb{F}(A)$.

**b): **

If $A=CSS_{2}^{A}(B)$ and $B\not=A$, then $ldim(ker(\gamma_{B}^{A}))<\delta_{2}(B)-\delta_{2}(A)$.

We need the Theorem for $B=C^{*}$ and $A=D^{*}$ where $C\subseteq D$ in $\mathcal{K}_{3}$. Then we have $\mathbb{F}(A)\in\mathcal{K}_{3}$. First we prove:

Lemma 5.6**.**

*Let $\langle E_{1},a\rangle=A$ be in $\mathcal{K}_{2}^{fin}$, where $a\in A_{1}\setminus E_{1}$. Let $F(A_{1})$ be the free 3-nilpotent graded Lie algebra over the vectorspace $A_{1}$. Furthermore $A=F(A_{1})^{*}/I$. Assume that $C_{1}\subseteq E_{1}\subseteq A_{1}$, $c_{1},\ldots,c_{n}$ is a vector space basis of $C_{1}$, such that $[c_{i},a]+\psi_{i}$ with $\psi_{i}\in F(E_{1})_{2}$ build a vector space basis of $I$ over $F(E_{1})^{*}_{2}$. Let $N_{2}(C)$ be $I\cap\langle C_{1}\rangle_{2}$.

If $n=1$, then $ldim(ker(\gamma_{E}^{A}))=0$. Otherwise*

[TABLE]

Proof.

By assumption $A=\langle A_{1}\rangle$ and $E=\langle E_{1}\rangle$ in $\mathcal{K}_{2}^{fin}$. Mainly we work in $F(A_{1})$ the free graded 3-nilpotent Lie algebra over $K$ freely generated by $A_{1}$. In $F(A_{1})$ we use the same notation for the $\tau^{-1}$-images of subsets and elements in $F(A_{1})^{*}$, as e.g. $I$ and $[c_{i},a]+\psi_{i}$.

Then $\mathbb{F}(A)=F(A_{1})/J$, where $J=\langle\langle\ I\rangle\rangle^{F(A_{1})}$. The non-zero elements of $ker(\gamma_{E}^{A})\subseteq\mathbb{F}(E_{1})_{3}$ have preimages $\mu\in J_{3}\cap\langle E_{1}\rangle^{F(A_{1})}_{3}$ that are not in $\langle\langle I\cap F(E_{1})\rangle\rangle^{F(E_{1})}$, where $F(E_{1})$ is considered as a subalgebra of $F(A_{1})$. W.l.o.g.

[TABLE]

where $e_{i}\in A_{1}$ and $\theta\in I_{2}\cap E_{2}$. Since cancellation of $[a,[c_{i},a]]$ is impossible, we have that $e_{i}\in E_{1}$. Furthermore $e_{i}\in C_{1}$. Otherwise $[e_{i},[c_{i},a]]$ cannot be killed by monomials $[a,[c_{i},e_{i}]]$. Hence

[TABLE]

We have $i\neq j$, since we cannot cancel $[c_{i},[c_{i},a]]$. Since all monomials with $a$ have to vanish, we get

[TABLE]

Then it follows that the $\theta^{\mu}$ are in $I_{2}\cap\langle C_{1}\rangle$. Let s be the linear dimension of $ker(\gamma_{E}^{A})$. If $n=1$, then there is no $\mu$ and $s=0$. If we have s many linearly independent elements in $ker(\gamma_{E}^{A})$, then we have s such $\mu$ as above linearly independent over $\langle\langle I_{2}\cap E_{2}\rangle\rangle^{F(E_{1})}$. Hence the

[TABLE]

are linearly independent and therefore also the $\theta^{\mu}$. By the definition of $\mathcal{K}_{2}$ and since $2\leq n$, we have $\delta_{2}(\langle C_{1}\rangle)\geq 2$. Therefore $s<n-1$ and $ldim(ker(\gamma_{E}^{A})=s<n-1=\delta_{2}(B)-\delta_{2}(A)$. ∎

Now we show Theorem 5.5

Proof.

First we show, that it is sufficient to consider $\langle B_{1}\rangle\subseteq\langle A_{1}\rangle$. We assume that the assertion is true in this case and $A\neq B$.

Case a) By Lemma 5.2 we have $\langle B_{1}\rangle\leq_{2}A$. Hence by assumption the homomorphism of $\mathbb{F}(\langle B_{1}\rangle)$ into $\mathbb{F}(\langle A_{1}\rangle)$ is an embedding. By Lemma 5.4 we can consider $\mathbb{F}(\langle A_{1}\rangle)$ as a substructure of $\mathbb{F}(A)$ .

This is the first step of an induction on $o-dim_{2}(B)$ to show Case a). We can assume that $B_{1}\neq\langle 0\rangle$, since n linearly independent elements in any $\mathbb{F}(A)_{2}$ generate a substructure $D$ with $D_{1}=\langle 0\rangle$, $D_{2}$ is a vector space with a basis of n elements, and $D_{3}=\langle 0\rangle$.

For the induction we consider $\langle Bb\rangle\leq A$ with $b\in A_{2}\setminus B$ and the canonical homomorphism $\gamma$ of $\mathbb{F}(B)$ into $\mathbb{F}(A)$ is an embedding. By Lemma 5.4 $\langle Bb\rangle=B\otimes\langle b\rangle$. We distingush the following:

1. i)

There are $c\in B_{1}$ and $a\in A_{1}$ such that $[c,a]=b$.

Then $\langle Ba\rangle\leq A$. By induction there is an embedding $\gamma^{+}$ of $\mathbb{F}(\langle Ba\rangle)$ into $\mathbb{F}(A)$ that extends $\gamma$. Furthermore we get $\langle Ba\rangle=\langle Bb\rangle\otimes_{\langle c,b\rangle}\langle c,a\rangle$. Then $\mathbb{F}(\langle b,c\rangle)$ can be considered as a substructure of $\mathbb{F}(\langle c,a\rangle)$ and of $\mathbb{F}(\langle Bb\rangle)$ since $\langle Bb\rangle=B\otimes\langle b\rangle$. We see easily $\mathbb{F}(\langle Ba\rangle)=\mathbb{F}(\langle Bb\rangle)\otimes_{\mathbb{F}(\langle bc\rangle)}\mathbb{F}(\langle c,a\rangle)$. Hence we obtain an embedding of $\mathbb{F}(\langle Bb\rangle)$ into $\mathbb{F}(\langle Ba\rangle)$ and then into $\mathbb{F}(A)$. 2. ii)

There are no $c\in B_{1}$ and $a\in A_{1}$ with $[c,a]=b$.

By assumption there is some $c\in B_{1}$. We fix some new element $a$ with $U_{1}(a)$, $[c,a]=b$, and define $B^{+}=\langle Bb\rangle\otimes_{\langle c,b\rangle}\langle a,c,b\rangle$ and $A^{+}=A\otimes_{\langle c,b\rangle}\langle a,c,b\rangle$.

Then $B^{+}\leq A^{+}$. By induction there is an embedding of $\mathbb{F}(B^{+})$ into $\mathbb{F}(A^{+})$. As above we can show that $\mathbb{F}(\langle Bb\rangle)$ and $\mathbb{F}(A)$ can be considered as substructures of $\mathbb{F}(B^{+})$ and $\mathbb{F}(A^{+})$ respectively.

Case b) Since $A=CSS_{2}(B)$ we have $A=\langle A_{1}B\rangle$. We assume $B=\langle B_{1}Y\rangle$, where $Y=\{y_{1},\ldots,y_{n}\}$ is a vector space basis of $B_{2}$ over $\langle B_{1}\rangle$. By induction we define the structure $\langle zy_{i}x_{i}\rangle$ by $U_{1}(x_{i})$, $U_{1}(z)$, and $[z,x_{i}]=y_{i}$. Furthermore let $C$ be $\otimes^{1\leq i\leq n}_{\langle z\rangle}\langle zx_{i}y_{i}\rangle$. We define $B^{+}=C\otimes_{\langle y_{1},\ldots,y_{n}\rangle}B$ and $A^{+}=C\otimes_{\langle y_{1},\ldots,y_{n}\rangle}A$. Then $B^{+}\subseteq A^{+}$, $\delta(B^{+})=\delta(B)+1$, and $\delta(A^{+})=\delta(A)+1$.

$B\leq\langle Bz\rangle\leq\ldots\leq\langle Bzx_{1}\ldots x_{i}\rangle$ and $A\leq\langle Az\rangle\leq\ldots\leq\langle Azx_{1}\ldots x_{i}\rangle$. Hence $B\leq B^{+}$ and $A\leq A^{+}$. By a) we can assume that $\mathbb{F}(B)\subseteq\mathbb{F}(B^{+})$ and $\mathbb{F}(A)\subseteq\mathbb{F}(A^{+})$.

We can apply the assumption since $B^{+}=\langle B^{+}_{1}\rangle$, $A^{+}=\langle A^{+}_{1}\rangle$, and $CSS^{A^{+}}(B^{+})=A^{+}$. Hence

[TABLE]

Now we assume $B=\langle B_{1}\rangle$ and $A=\langle A_{1}\rangle$. We use induction on $ldim(A_{1}/B_{1})$:

Case 1) $ldim(A_{1}/B_{1})=1$

We use Lemma 5.6. Let $B$ be $E$.

ad a) Since $B\leq_{2}A$, we have $n\leq 1$. By Lemma 5.6 $ldim(ker(\gamma_{B}^{A}))\leq 0$, as desired.

ad b) Again by Lemma 5.6 $ldim(ker(\gamma_{B}^{A}))<n-1=\delta_{2}(B)-\delta_{2}(A)$.

Now we distinguish two cases for the induction step.

Case 2) There is some $D=\langle D_{1}\rangle$ such that $B\subseteq D\subseteq A$, $B\neq D\neq A$, and $\delta_{2}(A)\leq\delta_{2}(D)\leq\delta_{2}(B)$.

ad a) Since $B\leq_{2}A$ we have $\delta_{2}(B)=\delta_{2}(D)=\delta_{2}(A)$. Then $B\leq_{2}D$ and $D\leq_{2}A$. By induction $\gamma_{B}^{D}$ and $\gamma_{D}^{A}$ are embeddings. Hence $\gamma_{B}^{A}$ is an embedding.

ad b) We choose $D_{1}$ minimal with the properties of Case 2). We have $A=CSS_{2}^{A}(D)$ and $\delta_{2}(A)<\delta_{2}(D)$. By the minimality of $D$ we get either $\delta_{2}(D)<\delta_{2}(B)$ and $CSS_{2}^{D}(B)=D$ or $\delta_{2}(D)=\delta_{2}(B)$ and $B\leq_{2}D$. By induction for a) and b) we have

[TABLE]

[TABLE]

Then

[TABLE]

Case 3) For all $D$ with $D=\langle D_{1}\rangle$ such that $B\subseteq D\subseteq A$, and $B\neq D\neq A$, $\delta_{2}(D)<\delta_{2}(A)$ or $\delta_{2}(B)<\delta_{2}(D)$.

Case 3.1 For some $D$ as above $\delta_{2}(D)<\delta_{2}(A)$. This is only possible in a). We choose such a $D_{1}$ maximal. Then $B\leq_{2}D$ and $D\leq_{2}A$. The assertion follows by induction.

Case 3.2) For all $D$ with $D=\langle D_{1}\rangle$ such that $B\subseteq D\subseteq A$, and $B\neq D\neq A$, we have $\delta_{2}(D)\geq\delta_{2}(A)$ and $\delta_{2}(B)<\delta_{2}(D)$.

If for some $D^{0}$ with $D^{0}=\langle D_{1}^{0}\rangle$ such that $B\subseteq D^{0}\subseteq A$, and $B\neq D^{0}\neq A$ we have $\delta_{2}(D^{0})=\delta_{2}(A)$, then we are in a) and $D^{0}\leq_{2}A$ . We apply induction.

Finally we can assume that for all $D$ with $D=\langle D_{1}\rangle$ such that $B\subseteq D\subseteq A$, and $B\neq D\neq A$ we have $\delta_{2}(A)<\delta_{2}(D)$ and $\delta_{2}(B)<\delta_{2}(D)$. Then $\delta_{2}(A)\leq\delta_{2}(B)$, since otherwise $\delta_{2}(Bd)\leq\delta_{2}(A)$ for every $d\in A_{1}\setminus B_{1}$. Note that $\delta_{2}(Bd)\leq\delta_{2}(B)+1$.

Now we choose $B_{1}\subseteq E_{1}$ and $a\in A_{1}\setminus E_{1}$ such that $\langle E_{1}a\rangle=A$. Hence by assumption $\delta_{2}(A)<\delta_{2}(E)$. We will show that $ldim(ker(\gamma_{B}^{A}))\leq max\{0,\delta_{2}(B)-\delta_{2}(A)-1\}$. Then a) and b) follow. Note that in a) $\delta_{2}(B)=\delta_{2}(A)$.

As in Lemma 5.6 we assume that $C_{1}\subseteq E_{1}$, $c_{1},\ldots,c_{n}$ is a vector space basis of $C_{1}$, such that $[c_{i},a]+\psi_{i}$ with $\psi_{i}\in F(E_{1})_{2}$ is a basis of $I$ over $F(E_{1})_{2}^{*}$, where $A=F(A_{1})^{*}/I$. Now we work in $F(A_{1})$. As above we use the same notations for subsets and elements of $F(X)$ as for their $\tau$-images in $F(X)^{*}$. Define $J=\langle\langle I\rangle\rangle^{F(A_{1})}$ as above. For every $D=\langle D_{1}\rangle$ with $D_{1}\subseteq A_{1}$ we use $N_{2}(D)$ for $I_{2}\cap\langle D_{1}\rangle_{2}$. W.l.o.g. we assume, that there is some $m$, such that $c_{1},\ldots,c_{m}\in B_{1}$ and $c_{m+1},\ldots,c_{n}$ are linearly independent over $B_{1}$. Furthermore let $c_{m+1},\ldots,c_{n},e_{1},\ldots,e_{r}$be a basis of $E_{1}$ over $B_{1}$. For $B_{1}$ we choose a basis that extends $c_{1},\ldots,c_{m}$.

First we assume that $0<r$. We order our basis of $E_{1}$, as given above, in such a way that $x<e_{r}$ for all $x\not=e_{r}$. The set of all $[x,y]$ with $y<x$ is a vector space basis of $\langle E_{1}\rangle_{2}$. Hence all monomials $[x,e_{r}]$, where $x\not=e_{r}$, are in this basis. We consider the occurence of monomials $[x,e_{r}]$ in the $\psi_{i}$. We show that they do not occure and apply induction. We consider the $[c_{i},a]+\psi_{i}$. After linear transformations we can assume w.l.o.g. that there are $1\leq l_{b}\leq m+1$, $m+1\leq l_{a}\leq n+1$, and $\chi_{i}=[x_{i},e_{r}]$ with $x_{i}$ from the vector basis of $E_{1}$ for $l_{b}\leq i\leq m$ and $l_{a}\leq i\leq n$ that are linearly independent such that :

**: **

In $\psi_{i}$ for $1\leq i<l_{b}$ no $\chi_{j}$ occures.

**: **

For $l_{b}\leq i\leq m$ $\chi_{i}$ occures in $\psi_{i}$ and not in $\psi_{j}$ for $j\not=i$..

**: **

In $\psi_{i}$ for $m+1\leq i<l_{a}$ no $\chi_{j}$ occures.

**: **

$\chi_{j}$ for $l_{a}\leq j\leq n$ occures in $\psi_{j}$ and does not occure in $\psi_{i}$ for $m+1\leq i\leq n$ with $i\neq j$.

As in Lemma 5.6 we consider the preimage $\mu$ in $F(A_{1})$ of an element in $ker(\gamma_{E}^{A})$. As in the proof of that lemma we get

[TABLE]

If $r_{i,j}^{\mu}\neq 0$ for some $l_{b}\leq i\leq m$, then $\chi_{i}$ cannot be canceled. Otherwise $l_{b}=m+1$. If in this case $r_{i,j}\neq 0$ for some $l_{a}\leq i\leq n$, then $\chi_{i}$ cannot be canceled. Hence if $\mu$ represents an element of $ker(\gamma_{B}^{A})$, then it is impossible that $e_{r}$ occures in some $\psi_{i}$ with $r_{i,j}^{\mu}\neq 0$ for some $j$. Let $E(-)$ be generated by $B_{1}$ and $c_{m+1},\ldots,c_{n},e_{1},\ldots,e_{r-1}$ and $A(-)=\langle E(-)_{1}a\rangle$. We have shown, that $ker(\gamma_{B}^{A})=ker(\gamma_{B}^{A(-)})$. By the final assumption of Case 3.2) $B\leq_{2}A(-)$ and by induction $ker(\gamma_{B}^{A(-)})=0$. The assertion is shown.

It remains the case where $E_{1}=B_{1},c_{m+1},\ldots,c_{n}$. By Lemma 5.6 $ldim(ker(\gamma_{E}^{A}))\leq k<n-1$, where $k=ldim(N_{2}(C))$.

Then $0<m$, since otherwise no $\mu$ as above can represent an element of $ker(\gamma_{B}^{A})$. First we assume that $2\leq m$.

Let $k_{b}$ be $ldim(N_{2}(c_{1},\ldots,c_{m}))$ and $k_{a}=k-k_{b}$. Since $B\leq_{2}E$ $\gamma_{B}^{E}$ is an embedding by induction. Hence in $\mathbb{F}(E)$ we have $ker(\gamma_{B}^{A})\subseteq(ker\gamma_{E}^{A})$. Since $E_{1}$ is generated by $c_{m+1},\ldots,c_{n}$ over $B_{1}$, we have

[TABLE]

[TABLE]

We will use:

[TABLE]

[TABLE]

and since $2\leq m$,

[TABLE]

It follows

[TABLE]

It remains the case $m=1$. We use the proof of Lemma 5.6 and show that

$ker(\gamma_{B}^{A})=\langle 0\rangle$.

The preimage of a nontrivial element in $ker(\gamma_{B}^{A})\subseteq\mathbb{F}(B)\subseteq\mathbb{F}(E)$ in $F(\langle A_{1}\rangle)$ has the form

[TABLE]

Then the image of $\mu$ in $\mathbb{F}(B)$ has the form $[c_{1},d]$, where $d\in\mathbb{F}(B)_{2}$. This is impossible since $\mathbb{F}(A)$ is $\mathcal{K}_{3}$. Therefore it has no homogeneous zero-divisors. ∎

Corollary 5.7**.**

Let $D$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$.

1. (1)

If $B\subseteq D$ , $A=CSS_{2}^{D}(B)$ , and $B\not=A$, then $\delta(A)<\delta(B)$. 2. (2)

For graded Lie algebras $M$ in $\mathcal{K}_{3}$ we can define $B\leq M$, if $\delta(B)\leq\delta(A)$ for all $B\subseteq A\subseteq M$.

Proof.

ad(1) We can assume $D=A$. Note that $o-dim_{3}(D)\leq o-dim(B)$. In $\mathbb{F}(D^{*})$ we have $\gamma_{B^{*}}^{D^{*}}(N_{3}(B))\subseteq N_{3}(D)$ and

[TABLE]

By Theorem 5.5 $ldim(ker(\gamma_{B^{*}}^{D^{*}}))<\delta_{2}(B)-\delta_{2}(D)$. Hence

[TABLE]

[TABLE]

[TABLE]

(2) follows from (1). ∎

Lemma 5.8**.**

Assume $A\leq^{n}M\in\mathcal{K}_{3}$ and $A\subseteq C\subseteq D$ with $o-dim(D^{*}/A^{*})\leq n$. Then $\delta(A)\leq\delta(C)$.

Proof.

We can assume w.l.o.g. that $D=\langle D_{1}D_{2}A_{3}\rangle$.

By Corollary 5.7 $\delta(CSS_{2}^{D}(C))\leq\delta(C)$. Hence w.l.o.g. $C\leq_{2}D$.

Then $\mathbb{F}(C^{*})$ can be considered as a subspace of $\mathbb{F}(D^{*})$. We have $C^{-}=\mathbb{F}(C^{*})/N_{C}$ and $D^{-}=\mathbb{F}(D^{*})/N_{D}$. Then $ldim(N_{C})\leq ldim(N_{D})$.

If $o-dim(C^{*}/A^{*})\leq n$, then we are done. Otherwise we show $\delta(A)\leq\delta(C)$.

Let $X=X_{2}^{C}(A)X_{2}^{C}(C)$ be a vector space basis of $C_{2}$ over $\langle C_{1}\rangle_{2}$, where $X_{2}^{C}(A)$ is a vector space basis of $A_{2}$ over $\langle C_{1}\rangle_{2}$ and $X_{2}^{C}(C)$ is a vector space basis of $C_{2}$ over $\langle C_{1}A_{2}\rangle_{2}$. Let $X_{3}^{C}$ be a vector space basis of $C_{3}$ over $C^{-}_{3}$. Now

[TABLE]

Furthermore

[TABLE]

where $X_{2}^{D}(A)$ is a vector space basis of $A_{2}$ over $\langle D_{1}\rangle_{2}$, $X_{2}^{D}(D)$ is a vector space basis of $D_{2}$ over $\langle D_{1}A_{2}\rangle_{2}$ and $X_{3}^{D}$ is a vector space basis of $D_{3}$ over $D^{-}_{3}$.

Then $\mid X_{3}^{C}\mid\geq\mid X^{D}_{3}\mid$ and $\mid X_{2}^{D}(A)\mid\leq\mid X_{2}^{C}(A)\mid$. By assumption

[TABLE]

Hence $\delta(D)\leq\delta(C)$.

∎

Corollary 5.9**.**

Assume $M\in\mathcal{K}_{3}$. $A\leq^{n}M$ if and only if $\delta(A)\leq\delta(C)$ for all $A\subseteq C\subseteq M$, that are contained in some $D\subseteq M$ with $o-dim(D^{*}/A^{*})\leq n$.

6. submodularity and amalgamation for $\mathcal{K}_{3}$

From now on we concentrat on 3-nilpotent graded Lie algebras.

Lemma 6.1**.**

We work in a 3-nilpotent graded Lie algebra $M\in\mathcal{K}_{3}$. Assume $C\cap A=B$ and $\langle CA\rangle^{*}=C^{*}\otimes_{B^{*}}A^{*}$, $\alpha$ is the the canonical homomorphism of $\mathbb{F}(B^{*})$ into $\mathbb{F}(A^{*})$, and $\gamma$ the canonical homomorphism of $\mathbb{F}(B^{*})$ into $\mathbb{F}(C^{*})$. We identify $\mathbb{F}(B^{*})/\langle ker(\alpha),ker(\gamma)\rangle$ with the isomorphic substructures in $\mathbb{F}(A^{*})/\alpha(ker(\gamma))$ and in $\mathbb{F}(C^{*})/\gamma(ker(\alpha))$. Then

[TABLE]

The isomorphism is given by the fact, that we can identify the ∗-images of both sides.

Proof.

If we apply ∗ to the right side, then the result is isomorphic to $\langle CA\rangle^{*}$. Hence there is a homomorphism $h$ from the left side onto the right side.

Conversely there are homomorphisms $f$ of $\mathbb{F}(C^{*})/\gamma(ker(\alpha))$ and $g$ of $\mathbb{F}(A^{*})/\alpha(ker(\gamma))$ into $\mathbb{F}(\langle AC\rangle^{*})$. $f$ and $g$ coincide on the intersection isomorphic to

$\mathbb{F}(B^{*})/\langle ker(\alpha),ker(\gamma)\rangle$. Hence there is a homomorphism $k$ of the right side onto the left side. Since the ∗-images $h^{*}=(k^{*})^{-1}$ are isomorphisms we get the assertion. Note that we have $\mathbb{F}(E^{*})=\langle\mathbb{F}(E^{*})_{1}\mathbb{F}(E^{*})_{2}\rangle^{\mathbb{F}(E^{*})}$. ∎

Corollary 6.2**.**

*We are in the setting of Lemma 6.1.

Let $X_{1}^{a}$ be an ordered vector space basis of $A_{1}$ over $B_{1}$ ,

$X_{1}^{c}$ be an ordered vector space basis of $C_{1}$ over $B_{1}$,

$X_{2}^{a}$ be a vector space basis of $A_{2}$ over $B_{2}$, and

$X_{2}^{c}$ be a vector space basis of $C_{2}$ over $B_{2}$. Then*

1. (1)

$\{[x,y]:x\in X_{1}^{c},y\in X_{1}^{a}\}$* is a vector space basis of $\mathbb{F}(\langle CA\rangle^{*})_{2}$ over $C_{2}+A_{2}$,* 2. (2)

[TABLE]

[TABLE]

[TABLE]

is a vectorspace basis of $\mathbb{F}(\langle CA\rangle^{*})_{3}$ over $\langle C_{1}C_{2}\rangle_{3}^{\mathbb{F}(\langle CA\rangle^{*})}+\langle A_{1}A_{2}\rangle_{3}^{\mathbb{F}(\langle CA\rangle^{*})}$.

The Corollary is a consequence of Lemma 6.1, Corollary 3.7, and Corollary 3.9.

Lemma 6.3**.**

Let $M$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$ and $A$ and $C$ are finite substructures of $M$ with $A\leq_{2}M$ and $C\leq_{2}M$. Then

[TABLE]

Proof.

Let $B$ be $A\cap C$. By Lemma 2.8 we have

**1): **

$\delta_{2}(\langle AC\rangle\leq\delta_{2}(A)+\delta_{2}(C)-\delta_{2}(B).$

We can assume w.l.o.g. that $M=\langle AC\rangle$. Let $M^{-}=\langle M_{1}M_{2}\rangle$ be isomorphic to $\mathbb{F}(M^{*})/N$, where $N\subseteq\mathbb{F}(M^{*})_{3}$. Furthermore we define

[TABLE]

Since $A\leq_{2}M$ and $C\leq_{2}M$, we have $A\cap C\leq_{2}M$ by Lemma 4.1. Hence we can consider $\mathbb{F}(A^{*}),\mathbb{F}(C^{*}),\mathbb{F}(A^{*}\cap C^{*})=\mathbb{F}(B^{*})$ as substrucures of $\mathbb{F}(M^{*})$

by Theorem 5.5.

Since $\mathbb{F}((B^{*})_{i}=\mathbb{F}(A^{*})_{i}\cap\mathbb{F}(C^{*})_{i}$ for $i=1,2$, there is an homomorphism of $\mathbb{F}(B^{*})$ onto $\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*})$. Hence $\mathbb{F}(B^{*})=\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*})$.

Under these assumptions we have $N(A)=N\cap\mathbb{F}(A^{*})_{3}$, $N(C)=N\cap\mathbb{F}(C^{*})_{3}$, and $N(B)=N\cap\mathbb{F}(B^{*})_{3}$.

Then $N(C)\cap N(A)=N(B)$. We get:

**2): **

$ldim(N(\langle AC\rangle))\geq ldim(N(A))+ldim(N(C))-ldim(N(B))+r.$

where $r$ is the linear dimension of all $\psi_{a}+\psi_{b}$ in $N(\langle AC\rangle)$ over $N(A)+N(C)$ with $\psi_{a}\in\mathbb{F}(A^{*})_{3}$ and $\psi_{b}\in\mathbb{F}(C^{*})_{3}$ .

Now we have to compute the $o-dimensions_{3}$. In $M_{3}$ we define:

**: **

$X_{3}^{0}(B)$ is a vector basis for $\langle C_{1}C_{2}\rangle_{3}\cap\langle A_{1}A_{2}\rangle_{3}$ over $\langle B_{1}B_{2}\rangle_{3}$.

**: **

$X_{3}^{A}(B)$ is a vector basis for $\langle A_{1}A_{2}\rangle_{3}\cap C_{3}$ over $\langle B_{1}B_{2}\rangle_{3}+\langle X_{3}^{0}(B)\rangle$.

**: **

$X_{3}^{C}(B)$ is a vector basis for $\langle C_{1}C_{2}\rangle_{3}\cap A_{3}$ over $\langle B_{1}B_{2}\rangle_{3}+\langle X_{3}^{0}(B)\rangle$.

**: **

$X_{3}^{1}(B)$ is a vector basis for $B_{3}\cap\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}$ over $\langle B_{1}B_{2}\rangle_{3}+\langle X_{3}^{0}(B)X_{3}^{A}(B)X_{3}^{C}(B)\rangle$.

**: **

$X_{3}^{2}(B)$ is a vector basis of $B_{3}$ over $\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}$.

Note that $X_{3}^{A}(B)\cup X_{3}^{C}(B)$ is linearly independent over $\langle B_{1}B_{2}\rangle_{2}+\langle X_{3}^{0}(B)\rangle_{3}$. Now $B_{3}$ has the following vector space basis $X_{3}(B)$ over $\langle B_{1}B_{2}\rangle_{3}$:

[TABLE]

Let $X_{3}^{1}(A)$ and $X_{3}^{1}(C)$ be choosen such that

**: **

$X_{3}^{C}(B)X_{3}^{1}(B)X_{3}^{1}(A)$ is a vector basis of $A_{3}\cap\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}$ over $\langle A_{1}A_{2}\rangle_{2}$.

**: **

$X_{3}^{A}(B)X_{3}^{1}(B)X_{3}^{1}(C)$ is a vector basis of $C_{3}\cap\langle A_{1}A_{2}C_{1}C_{2}\rangle_{2}$ over $\langle C_{1}C_{2}\rangle_{2}$.

Choose $X_{3}^{2}(A)$ and $X_{3}^{2}(C)$ such that

**: **

$X_{3}(A)=X_{3}^{C}(B)X_{3}^{1}(B)X_{3}^{2}(B)X_{3}^{1}(A)X_{3}^{2}(A)$ is a vector basis of $A_{3}$ over $\langle A_{1}A_{2}\rangle_{3}$.

**: **

$X_{3}(C)=X_{3}^{A}(B)X_{3}^{1}(B)X_{3}^{2}(B)X_{3}^{1}(C)X_{3}^{2}(C)$ is a vector basis of $C_{3}$ over $\langle C_{1}C_{2}\rangle_{3}$.

Then $X_{3}(AC)=X_{3}^{2}(B)X_{3}^{2}(A)X_{3}^{2}(C)$ is a vector basis of $\langle AC\rangle_{3}$ over $\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}$.

It follows:

**3): **

$o-dim_{3}(A)+o-dim_{3}(C)-o-dim_{3}(B)=$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $\mid X_{3}^{0}(B)\mid=r$, 1), 2), and 3) imply

[TABLE]

∎

If we repeat the proof above without the assumptions $C\leq_{2}M$ and $A\leq_{2}M$, then we obtain that $\delta(A)+\delta(C)-\delta(B)-\delta(\langle AC\rangle)$ is the sum of the following summands:

1. $\Sigma 1$

$\delta_{2}(A)+\delta_{2}(C)-\delta_{2}(B)-\delta_{2}(\langle AC\rangle)$ 2. $\Sigma 2$

$\mid X_{3}^{1}(B)X_{3}^{1}(A)X_{3}^{1}(C)\mid$ 3. $\Sigma 3$

$ldim(N)+ldim(N(B))-ldim(N(C))-ldim(N(A))-\mid X_{3}^{0}(B)\mid$.

In this general case we have $0\leq\Sigma 1$ by Lemma 2.8. If $\Sigma 3\geq 0$, then we have subadditivity.

We use this remark to get the following:

Lemma 6.4**.**

*Let $M$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$, $M=\langle CA\rangle$, $C\cap A=B$, and $B\leq_{2}A$. We choose $X_{B}$ as a maximal subset of $B^{*}_{2}\cap\langle C_{1}A_{1}\rangle_{2}$ linearly independent over $\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2}$. Then let $X_{C}$ be maximal subset of $C^{*}_{2}\cap\langle C_{1}A_{1}\rangle_{2}$ that is linearly independent over $\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2}+\langle X_{B}\rangle_{2}$, and $X_{A}$ be maximal subset of $A^{*}_{2}\cap\langle C_{1}A_{1}\rangle_{2}$ that is linearly independent over $\langle C_{1}\rangle_{2}+\langle A_{1}\rangle_{2}+\langle X_{B}\rangle_{2}$.

Furthermore there are $D\subseteq C^{*}$ $H\subseteq B^{*}$, $K\subseteq A^{*}$ such that*

1. (1)

$B^{*}=\langle HX_{B}\rangle$, $C^{*}=\langle DX_{B}X_{C}\rangle$, $A^{*}=\langle KX_{B}X_{A}\rangle$, 2. (2)

$X_{B}$* is linearly independent over $H_{2}$,*

$X_{B}X_{C}$* is linearly independent over $D_{2}$,*

$X_{B}X_{A}$* is linearly independent over $K_{2}$, and* 3. (3)

$\langle C^{*}A^{*}\rangle=\langle DK\rangle=D\otimes_{H}K.$**

Then

1. (1)

$\delta(\langle CA\rangle)\leq\delta(C)+\delta(A)-\delta(B)$. 2. (2)

If $\mid X_{C}X_{B}X_{A}\mid>0$, then $\delta(\langle CA\rangle)<\delta(C)+\delta(A)-\delta(B)$. 3. (3)

If $C\leq\langle CA\rangle$, then $B\leq A$. 4. (4)

If $C^{*}=D$, $B^{*}=H$, $A^{*}=K$, $B\leq A$, and $\langle CA\rangle=C\otimes_{B}A$, then $\delta(\langle CA\rangle)=\delta(C)+\delta(A)-\delta(B)$ and $C\leq\langle CA\rangle$. 5. (5)

If $C^{*}=D$ , $B^{*}=H$, $A^{*}=K$, $B\leq A$, and $\delta(\langle CA\rangle)=\delta(C)+\delta(A)-\delta(B)$, then $\langle CA\rangle=C\otimes_{B}A$.

Proof.

Note that $X_{B}X_{C}X_{A}$ is linearly independent over $D_{2}+K_{2}$.

ad (1) We use the remark below Lemma 6.3. We have only to show that $0\leq\Sigma 3$. By Lemma 6.1 we get

[TABLE]

where $\gamma$ is the homomorphism of $\mathbb{F}(H)$ into $\mathbb{F}(D)$. Note that $B\leq_{2}A$ implies $\langle HX_{B}\rangle\leq_{2}\langle KX_{B}\rangle$. By Lemma 5.2 we get $H\leq_{2}K$. Therefore we can consider $\mathbb{F}(H)$ as a substructure of $\mathbb{F}(K)$ and $\mathbb{F}(B^{*})$ as a substructure of $\mathbb{F}(A^{*})$ by Theorem 5.5. By Theorem 4.2 $D\leq_{2}\langle DK\rangle$. Hence we consider $\mathbb{F}(D)$ as a substructure of $\mathbb{F}(\langle DK\rangle)$. Furthermore we consider substructures of $M^{*}$ as subspaces of $\mathbb{F}(M^{*})$:

We can rewrite $(\otimes)$ as

[TABLE]

where $\langle H\rangle^{\mathbb{F}(\langle DK\rangle}\cong\mathbb{F}(H)/ker(\gamma)$ and $\langle K\rangle^{\mathbb{F}(\langle DK\rangle)}\cong\mathbb{F}(K)/ker(\gamma)$.

By assumption $X_{B}X_{C}X_{A}$ is also linearly independent over $D_{2}+K_{2}$. Hence by Lemma 5.4 and ($\otimes$) we get that

($\otimes\otimes$)

[TABLE]

where $\langle HX_{B}\rangle^{\mathbb{F}(\langle DK\rangle)}=\langle H\rangle^{\mathbb{F}(\langle DK\rangle)}\otimes\langle X_{B}\rangle^{\mathbb{F}(\langle DK\rangle)}=\langle B^{*}\rangle^{\mathbb{F}(\langle DK\rangle)}\cong\mathbb{F}(B^{*})/ker(\gamma)$,

$\langle DX_{B}X_{C}\rangle^{\mathbb{F}(\langle DK\rangle)}=\langle D\rangle^{\mathbb{F}(\langle DK\rangle)}\otimes\langle X_{B}X_{C}\rangle^{\mathbb{F}(\langle DK\rangle)}=\langle C^{*}\rangle^{\mathbb{F}(\langle DK\rangle)}\cong\mathbb{F}(C^{*})$,

$\langle KX_{B}X_{A}\rangle^{\mathbb{F}(\langle DK\rangle)}=\langle K\rangle^{\mathbb{F}(\langle DK\rangle)}\otimes\langle X_{B}X_{A}\rangle^{\mathbb{F}(\langle DK\rangle)}=\langle A^{*}\rangle^{\mathbb{F}(\langle DK\rangle)}\cong\mathbb{F}(A^{*})/ker(\gamma)$.

As in the proof of Lemma 6.3 we define $\langle CA\rangle^{-}=\mathbb{F}(\langle C^{*}A^{*}\rangle)/N$, $C^{-}=\mathbb{F}(C^{*})/N(C)$, $B^{-}=\mathbb{F}(B^{*})/N(B)$, and $A^{-}=\mathbb{F}(A^{*})/N(A)$. By $(\otimes\otimes)$

[TABLE]

where $M(B)$ is the image of $N(B)/ker(\gamma)$ in $\langle HX_{B}\rangle^{\mathbb{F}(\langle DK\rangle)}_{3}$ ,

$M(C)$ is the image of $N(C)$ in $\langle DX_{B}X_{C}\rangle^{\mathbb{F}(\langle DK\rangle)}_{3}$,

$M(A)$ is the image of $N(A)/ker(\gamma)$ in $\langle KX_{B}X_{A}\rangle^{\mathbb{F}(\langle DK\rangle)}_{3}$, and

$U$ is generated by a maximal number $r=\mid X_{3}^{0}(B)\mid$ of $\psi_{C}+\psi_{A}\in N$, where the $\psi_{C}\in\langle C_{1}C_{2}X_{B}X_{C}\rangle_{3}$ are linearly independent over $M(C)_{3}$, and the $\psi_{A}\in\langle K_{1}K_{2}X_{B}X_{A}\rangle_{3}$ are linearly independent over $M(A)_{3}$.

Note that $ldim(N(B)\cap ker(\gamma))=ldim(N(A)\cap ker(\gamma))=s$. Since s will be canceled we get 2) as above:

[TABLE]

Hence $0\leq\Sigma 3$.

ad (2) If $\mid X_{C}X_{B}X_{A}\mid>0$, then $\Sigma 1>0$

ad (3) We have $B\leq_{2}A$ by assumption. Then (1) and $C\leq\langle CA\rangle$ imply $0\leq\delta(A/C)\leq\delta(A/B)$.

To prove the assertion choose $B\subseteq E\subseteq A$. We have to show that $\langle CE\rangle$ satisfies the assumptions of the Lemma. We have $B\leq_{2}E$. We choose $X_{E}$ as a subset of $\langle C_{1}E_{1}\rangle\cap E_{2}$ maximal linearly independent over $\langle C_{1}\rangle_{2}+\langle E_{1}\rangle_{2}+\langle X_{B}\rangle$. Note that we work in $M^{*}$. Let $K(E)$ be $K\cap E^{*}$. Then we have $E_{1}=K(E)_{1}$ and $\langle DK(E)\rangle=D\otimes_{H}K(E)$ by Mon.

Choose $Y\subseteq E_{2}$ maximal linearly independent over $\langle K(E)_{2}X_{B}X_{E}\rangle$. and define $K_{E}=\langle K(E)Y\rangle$. Then $\langle K_{E}X_{B}X_{E}\rangle=E^{*}$ and also

[TABLE]

Hence the assumptions of the Lemma are fulfilled and we get by (1) $\delta(\langle CE\rangle)\leq\delta(C)+\delta(E)-\delta(B)$. Since $C\leq\langle CE\rangle$ we get the assertion for $E$.

ad (4) As in (1) we use the remark below the proof of Lemma 6.3. We have $\langle C^{*}A^{*}\rangle=C^{*}\otimes_{B^{*}}A^{*}$. By Corollary 3.8 this implies $\delta_{2}(\langle CA\rangle)=\delta_{2}(C)+\delta_{2}(A)-\delta_{2}(B)$. This means $\Sigma 1=0$. By Corollary 3.9 we get $\mid X_{3}^{1}(B)X_{3}^{1}(C)X_{3}^{1}(A)\mid=0$. Hence $\Sigma 2=0$.

As in the proof of (1) we have

[TABLE]

where $\langle C^{*}\rangle^{\mathbb{F}(\langle C^{*}A^{*}\rangle)}\cong\mathbb{F}(C^{*})$, since $C^{*}\leq_{2}C^{*}\otimes_{B^{*}}A^{*}$,

$\langle B^{*}\rangle^{\mathbb{F}(\langle C^{*}A^{*}\rangle)}\cong\mathbb{F}(B^{*})/ker(\gamma)$, and

$\langle A^{*}\rangle^{\mathbb{F}(\langle C^{*}A^{*}\rangle)}\cong\mathbb{F}(A^{*})/ker(\gamma)$, where $\gamma$ is the homomorphism of $\mathbb{F}(B^{*})$ into $\mathbb{F}(C^{*})$ and w.l.o.g. $\mathbb{F}(B^{*})\subseteq\mathbb{F}(A^{*})$ by $B^{*}\leq_{2}A^{*}$.

Since $\langle CA\rangle=C\otimes_{B}A$ we get similarly as in (1)

[TABLE]

where $M(C)$, $M(B)$, and $M(A)$ are the preimages of $N(C)$, $N(B)$, and $N(A)$ respectively in $\mathbb{F}(\langle C^{*}A^{*}\rangle)$. $U$ is generated by $r=\mid X_{3}^{0}(B)\mid$ many $\psi_{C}+\psi_{A}\in N$, where the $\psi_{C}\in\langle C_{1}C_{2}\rangle_{3}$ are linearly independent over $M(C)$, and the $\psi_{A}\in\langle A_{1}A_{2}\rangle_{3}$ are linearly independent over $M(A)$. Then $ldim(N(C))=ldim(M(C))$, $ldim(N(B))=ldim(M(B))+s$, and $ldim(N(A))=ldim(M(A))+s$, with $s=ldim(ker(\gamma)\cap N(B))=ldim(ker(\gamma)\cap N(A))$.

Hence

[TABLE]

With other words $\Sigma 3=0$. We obtain $\delta(\langle CA\rangle)=\delta(C)+\delta(A)-\delta(B)$.

We use this to show $C\leq\langle CA\rangle$. For $C\subseteq E\subseteq C\otimes_{B}A$ we have to show $\delta(C)\leq\delta(E)$. By Mon $\langle C(E\cap A)\rangle=C\otimes_{B}(E\cap A)$. Then $\delta(C\otimes_{B}(A\cap E))\leq\delta(E)$ by Corollary 3.9. Hence it is suficient to consider the case $E=C\otimes_{B}(E\cap A)$. Furthermore $B\leq A\cap E$. Hence we can assume that w.l.o.g. $E=\langle CA\rangle$.

It follows $\delta(C)\leq\delta(\langle CA\rangle)$, since $B\leq A$ by assumption.

ad (5) We use again the remark below Lemma 6.3. By$\langle C^{*}A^{*}\rangle=C^{*}\otimes_{B^{*}}A^{*}$ we get $\Sigma 1=0$. From the proof of (1) we know, that $\Sigma 3\geq 0$. Hence by assumption $\Sigma 2=0$ and $\Sigma 3=0$. As above by Lemma 6.1 we get an isomorphism

[TABLE]

where $\gamma$ is the homomorphism of $\mathbb{F}(B^{*})$ into $\mathbb{F}(C^{*})$. Note that $B\leq_{2}A$ implies $\mathbb{F}(B^{*})\subseteq\mathbb{F}(A^{*})$. By Theorem 4.2 $C\leq_{2}\langle AC\rangle$. Hence $\mathbb{F}(C^{*})\subseteq\mathbb{F}(\langle AC\rangle^{*})$.

As in the proof of Lemma 6.3 we define $\langle CA\rangle^{-}=\mathbb{F}(\langle C^{*}A^{*}\rangle)/N$, $C^{-}=\mathbb{F}(C^{*})/N(C)$, $B^{-}=\mathbb{F}(B^{*})/N(B)$, and $A^{-}=\mathbb{F}(A^{*})/N(A)$. By $(\otimes)$

[TABLE]

where $U$ is generated by $r=\mid X_{3}^{0}(B)\mid$ many $\psi_{C}+\psi_{A}$, where $\psi_{C}\in\langle C_{1}C_{2}\rangle_{3}$ and $\psi_{A}\in\langle A_{1}A_{2}\rangle_{3}$, that are linearly independent over $N(C)+N(A)$. Note that $ldim(N(B)\cap ker(\gamma))=ldim(N(A)\cap ker(\gamma))=s$. Since s will be canceled we get :

[TABLE]

since $0=\Sigma 3$.

Then

• (=)

$f(N)=N(C)\oplus_{N(B)/ker(\gamma)}(N(A)/ker(\gamma))\oplus U.$

Assume we have homomorphisms $j$ of $C$ into $E$ and $g$ of $A$ into $E$, that coinside on $B$.

There are homomorphisms $j^{+}$ of $\mathbb{F}(C^{*})$ into $E$ and $g^{+}$ of $\mathbb{F}(A^{*})/ker(\gamma)$ into $E$, that coinside on $\mathbb{F}(B^{*})/ker(\gamma)$.

By $(\otimes)$ there is an homomorphism $h^{+}$ of $\mathbb{F}(\langle C^{*}A^{*}\rangle)$ into $E$.

By $(=)$ and $\Sigma 2=0$ we get an homomorphism $h^{-}$ of $\langle C^{-}A^{-}\rangle$ into $E$.

$h^{-}$ can be extended to a homomorphism $h$ of $\langle CA\rangle$ into $E$ . Hence $\langle CA\rangle=C\otimes_{B}A$. ∎

Lemma 6.5**.**

Assume $D=C\otimes_{B}A$ and $A=\langle B\bar{a}\rangle$, where $\bar{a}$ is a generating o-system of $A$ over $B$. Then we get:

1. (1)

$D^{*}=C^{*}\otimes_{B^{*}}A^{*}$** 2. (2)

If $E\subseteq B$, $D=C\otimes_{E}\langle E\bar{a}\rangle$, and $B\leq A$, then $C\leq D$ and $E\leq\langle E\bar{a}\rangle$.

Proof.

ad(1) This is a consequence of the Lemmas 3.9 and 3.8.

ad(2) By Lemma 5.7 we have $B^{*}\leq A^{*}$. By (1) $D^{*}=C^{*}\otimes_{E^{*}}\langle E\bar{a}\rangle^{*}$. By Lemma 6.4(4) we have $C\leq D$. By Lemma 4.1 $B^{*}\leq A^{*}$ implies $E\leq_{2}\langle E\bar{a}\rangle$. Then by Lemma 6.4 (3) we get $E\leq\langle E\bar{a}\rangle$. ∎

As in Definition 4.6 we define

Definition 6.6**.**

For $A,C,N\subseteq M\in\mathcal{K}_{3}$:

1. (1)

$\delta(A/C)=\delta\langle AC\rangle-\delta(C)$. 2. (2)

$\delta(A/N)=min\{\delta(\langle AC\rangle)-\delta(C):C\subseteq N\}$.

From the end of the proof of Lemma 6.3 and Corollary 2.9 we get

Corollary 6.7**.**

Let $M$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$ and $A$ and $C$ are 2-strong substructures. Then $\delta(A/C)=\delta(A/A\cap C)$ if and only if $\delta_{2}(A/C)=\delta_{2}(A/A\cap C)$, $A_{3}$ and $C_{3}$ do not contain any elements from $\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}\setminus\langle A_{1}A_{2}\rangle_{3}+\langle C_{1}C_{2}\rangle_{3}$ and if $N$ is choosen as in the proof of 6.3, then $N\subseteq\mathbb{F}(A^{*})_{3}+\mathbb{F}(C^{*})_{3}$.

Corollary 6.8**.**

Let $M$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$ and $A$ and $C$ are 2-strong substructures. Then $\delta(A/C)=\delta(A/A\cap C)$ if and only if $\langle AC\rangle=A\otimes_{A\cap C}C$.

Proof.

Let $B$ be $A\cap C$. Assume $\langle AC\rangle=A\otimes_{A\cap C}C$. Then $\langle A^{*}C^{*}\rangle=A^{*}\otimes_{A^{*}\cap C^{*}}C^{*}$. By Corollary 3.8 we have $\delta_{2}(A/C)=\delta_{2}(A/B)$. By Corollary 3.9 $A_{3}$ and $C_{3}$ do not contain any elements from $\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}\setminus\langle A_{1}A_{2}\rangle_{3}+\langle C_{1}C_{2}\rangle_{3}$. Let $N$ be chosen as in the proof of 6.3 .Then $N\subseteq\mathbb{F}(A^{*})_{3}+\mathbb{F}(C^{*})_{3}$. Hence $\delta(A/C)=\delta(A/A\cap C)$ by Corollary 6.7.

For the other direction we use the other direction of Corollary 6.7. Then $\delta_{2}(A/C)=\delta_{2}(A/B)$ implies $\langle A^{*}C^{*}\rangle=A^{*}\otimes_{A^{*}\cap C^{*}}C^{*}$ by Corollary 3.8. By assumption $A,C$ are 2-strong in $\langle AC\rangle$. Then $B$ is also 2-strong in $A$, $C$, and $\langle AC\rangle$. Hence we can consider $\mathbb{F}(A^{*}),\mathbb{F}(B^{*}),\mathbb{F}(C^{*})$ as subspaces of $\mathbb{F}(\langle A^{*}C^{*}\rangle)$. $\mathbb{F}(A^{*}\cap C^{*})\subseteq\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*})$. Since $(\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*}))^{*}=A^{*}\cap C^{*}$ , there is a homomorphism of $\mathbb{F}(A^{*}\cap C^{*})$ onto $\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*})$. Hence it is an isomorhism. We get equality and

[TABLE]

using the definiton of $\mathbb{F}$ and of the free amalgam. We choose $N\subseteq\mathbb{F}(\langle A^{*}C^{*}\rangle)_{3}$ such that $\langle A_{1}A_{2}C_{1}C_{2}\rangle=\mathbb{F}(\langle A^{*}C^{*}\rangle)/N$.

Let $f$ and $g$ be homomorphisms of $A$ and $C$ respectively into $E$, such that $f$ and $g$ coincide on $B$. Then there are homomrphisms $f^{+}$ of $\mathbb{F}(A^{*})$ into $E$ and $g^{+}$ of $\mathbb{F}(C^{*})$ into $E$ that coinside on $\mathbb{F}(B^{*})$. Since we have a free amalgam we get $h^{+}$ of $\mathbb{F}(A^{*})\otimes_{\mathbb{F}(B^{*})}\mathbb{F}(C^{*})$ into $E$. By Corollary 6.7 $N\subseteq\mathbb{F}(A^{*})_{3}+\mathbb{F}(C^{*})_{3}$ and $A_{3}$ and $C_{3}$ do not contain any elements from $\langle A_{1}A_{2}C_{1}C_{2}\rangle_{3}\setminus\langle A_{1}A_{2}\rangle_{3}+\langle C_{1}C_{2}\rangle_{3}$. Hence we get a homomorphism $h$ from $\langle A_{1}A_{2}C_{1}C_{2}\rangle$ into $E$ that extends $f$ and $g$. Then $h$ can be extended to $\langle AC\rangle$. Hence $\langle AC\rangle=A\otimes_{A\cap C}C$. ∎

By Lemma 2.11, Lemma 4.1, Corollary 5.7 and Lemma 6.3 above we get (1), (2) and (3) in the next Corollary.

Corollary 6.9**.**

Let $M$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$. Then for all substructures $A,C,E$ of $M$, where $A$ and $C$ are finite the following is true:

1. (1)

If $C\leq M$ and $E\leq_{2}M$, then $E\cap C\leq E$. 2. (2)

If $A\leq C\leq M$, then $A\leq M$. 3. (3)

If $A,C\leq M$, then $A\cap C\leq M$. 4. (4)

If $A\leq C\leq^{k}M$, then $A\leq^{k}M$. 5. (5)

If $B\leq^{k}M$, $C\leq^{k}M$, $o-dim(B^{*})<k$, and $o-dim(C^{*})<k$, then $B\cap C\leq^{k}M$.

Proof.

To prove (4), consider $A\subseteq E\subseteq M$ with $o-dim(E^{*}/A^{*})\leq k$. Then $o-dim(\langle CE\rangle^{*}/C^{*})\leq k$. By Corollary 5.9 we get $C\leq\langle CE\rangle$. By (2) we have $A\leq\langle CE\rangle$. Hence $\delta(A)\leq\delta(E)$.

ad(5): By Corollary 5.9 and the assumptions we get, that $B\leq\langle BC\rangle$ and $C\leq\langle BC\rangle$. Hence by (3) $B\cap C\leq\langle BC\rangle$ and $B\cap C\leq B$. Now assume $B\cap C\subseteq E\subseteq M$ and $o-dim(E^{*}/B^{*}\cap C^{*})<k$. Then $o-dim(\langle E^{*}B^{*}\rangle/B^{*})<k$. By Corollary 5.9 $B\leq\langle EB\rangle$. By $B\cap C\leq B$ we get $B\cap C\leq\langle EB\rangle$. Hence $\delta(B\cap C)\leq\delta(E)$, as desired. ∎

The Corollary allows the following definition.

Definition 6.10**.**

If $A$ is a finite substructure of $M$ in $\mathcal{K}_{3}$, then there exists a unique minimal $C$ with $A\subseteq C\leq M$. We define $C=CSS(A)$ - the selfsufficient closure.

Let $A^{-}$ be $\langle A_{1}A_{2}\rangle$

Lemma 6.11**.**

Let $A$ be a finite substructure $M\in\mathcal{K}_{3}$.

1. (1)

$CSS(A^{-})=CSS(A^{-})^{-}$* and $CSS(A^{-})\subseteq CSS(A)$.* 2. (2)

Either $CSS(A)=\langle CSS(A^{-}),A\rangle$ or there is some $C$ such that $CSS(A^{-})\leq C\leq M$, $C=C^{-}$, $CSS(A)=\langle CA\rangle$, and

[TABLE] 3. (3)

$CSS_{2}(A)\subseteq CSS(A)$. 4. (4)

If $B\subseteq A$, then $CSS(B)\subseteq CSS(A)$.

Proof.

ad (1) For every substructure $C\subseteq M$ we have $\delta(C)=\delta(C^{-})+o-dim_{3}(C)$.

By Lemma 6.3

[TABLE]

If $CSS(A^{-})\not\subseteq CSS(A)$, then $\delta(CSS(A^{-}))-\delta(CSS(A^{-})\cap CSS(A))<0$, and therefore $\delta(\langle CSS(A)CSS(A^{-})\rangle)-\delta(CSS(A))<0$,a contradiction.

ad(2) If $CSS(A)\not=\langle CSS(A^{-}),A\rangle$, then there is some proper strong extension $C$ of $CSS(A^{-})$ with $C=C^{-}$, $CSS(A)=\langle C,A\rangle$, and the inequation in (2) is true.

ad (3) By Lemma 6.3 we have

[TABLE]

If $CSS_{2}(A)\nsubseteq CSS(A)$, then $CSS_{2}(A)\not=A$ and $\delta_{2}(CSS_{2}(A))-\delta_{2}(CSS(A)\cap CSS_{2}(A))<0$. Hence $\delta(\langle CSS(A)CSS_{2}(A)\rangle)-\delta(CSS(A))<0$, a contradiction.

ad (4) This is clear by definition. ∎

In the following Lemma we summarize facts we have already proved and used:

Lemma 6.12**.**

Assume $A,B,C\subseteq M\in\mathcal{K}_{3}$ , $A,C\leq_{2}M$, and $A\cap C=B$. W.l.o.g. we use $\subseteq$ instead of the corresponding embeddings.

1. (1)

$\mathbb{F}(B^{*})\subseteq\mathbb{F}(A^{*})\subseteq\mathbb{F}(M^{*})$. 2. (2)

$\mathbb{F}(B^{*})\subseteq\mathbb{F}(C^{*})\subseteq\mathbb{F}(M^{*})$. 3. (3)

$\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*})=\mathbb{F}(B^{*})$. 4. (4)

$If\delta_{2}(A^{*}/C^{*})=\delta_{2}(A^{*}/B^{*})$, then $\langle A^{*}C^{*}\rangle=A^{*}\otimes_{B^{*}}C^{*}$ and $\mathbb{F}(\langle A^{*}C^{*}\rangle)=\mathbb{F}(A^{*})\otimes_{\mathbb{F}(B^{*})}\mathbb{F}(C^{*})$ 5. (5)

$\delta(A/C)=\delta(A/B)$* implies $\delta_{2}(A/C)=\delta_{2}(A/B)$ and $\langle AC\rangle=A\otimes_{B}C$.* 6. (6)

In general we have $\delta_{2}(A/C)\leq\delta_{2}(A/B)$ and $\delta(A/C)\leq\delta(A/B)$. Furthermore $\delta_{2}(A/C)<\delta_{2}(A/B)$ implies $\delta(A/C)<\delta(A/B)$.

Proof.

ad (1),(2) By Lemma 4.1 we have $B\leq_{2}A\leq_{2}M$ and $B\leq_{2}C\leq_{2}M$. By Theorem 5.5 we get the assertion.

ad (3) $\mathbb{F}(A^{*}\cap C^{*})_{i}=(\mathbb{F}(A^{*})\cap\mathbb{F}(C^{*}))_{i}=(A\cap C)_{i}$ for $i=1,2$.

ad(4) The first assertion follows from Corollary 2.9. Then we apply Lemma 6.1. By our assumption $\alpha$ and $\gamma$ are embeddings.

ad(5) It follows from Corollary 6.7 and 6.8.

ad(6) Here we have the subaddivity of $\delta_{2}$ and $\delta_{3}$. The last assertion follows from (5). ∎

Lemma 6.13**.**

Let $M$ be a 3-nilpotent graded Lie algebra in $\mathcal{K}_{3}$. For all $A\leq^{2n+n^{2}}C\leq M$ we have $A\leq^{n}M$.

Proof.

We consider any $A\subseteq E\subseteq M$, such that $o-dim(E^{*}/A^{*})\leq n$. By $C\leq\langle CE\rangle$ we get $\delta_{2}(E^{*}/C^{*})\geq 0$. Furthermore $o-dim_{1}(E^{*}/C^{*})\leq n$. By Lemma 4.8 we get:

Let $B$ be $E\cap C$. Then there exists $D$, $H$, $K$ in $M^{*}$, and $X_{C}X_{B}X_{E}$, such that

1. (1)

$H\subseteq D\subseteq C^{*}$, $B_{1}\subseteq H$, $D_{1}=C_{1}$, and $o-dim(H/B^{*})\leq n^{2}+n$, 2. (2)

$H\subseteq K\subseteq\langle HE^{*}\rangle$, $K_{1}=\langle HE^{*}\rangle_{1}$, 3. (3)

$\langle C^{*}E^{*}\rangle=\langle DK\rangle=D\otimes_{H}\langle K\rangle$, 4. (4)

$X_{C}X_{B}X_{A}\subseteq\langle C_{1}A_{1}\rangle_{2}$ is linearly independent over $\langle D_{1}\rangle_{2}+\langle K_{1}\rangle_{2}$, $X_{B}\subseteq B_{2}$, $\langle HX_{B}\rangle=\langle HB^{*}\rangle$, 5. (5)

$X_{C}\subseteq C_{2}$, $\langle DX_{B}X_{C}\rangle=C^{*}$, and 6. (6)

$X_{E}\subseteq E_{2}$, $\langle KX_{B}X_{E}\rangle=\langle HE^{*}\rangle$.

Now we can apply Lemma 6.4(3) to $C$, $\langle H_{1}H_{2}E\rangle=E_{H}$, and $C\cap E_{H}$ and obtain that $C\cap E_{H}\leq E$. Hence $\delta(C\cap E_{H})\leq\delta(E)$. Note that

$C\cap E_{H}=\langle C\cap E,H_{1},H_{2}\rangle=\langle B,H_{1},H_{2}\rangle$. Furthermore in $M^{*}$ holds

[TABLE]

Since $A\leq^{2n+n^{2}}C$, we have $\delta(A)\leq\delta(C\cap\langle E_{H}\rangle)\leq\delta(E)$. ∎

Lemma 6.14**.**

Assume that $A\subseteq E\subseteq D=C\otimes_{B}A$. Then $\delta((E\cap C)\otimes_{B}A)\leq\delta(E)$.

Proof.

By Mon $E^{0}=\langle(E\cap C)A\rangle=(E\cap C)\otimes_{B}A$. Since $A\subseteq E$ we have $E_{1}=E^{0}_{1}=\langle(E\cap C)_{1}A_{1}\rangle^{lin}$. $(E^{0})^{*}=\langle E_{1}X\rangle^{*}$, where $X\subseteq E^{0}_{2}$ linearly independent over $\langle E_{1}\rangle_{2}$. Next we have $E^{*}=\langle E_{1}XY\rangle^{*}$, where $Y\subseteq E_{2}$ is linearly independent over $(E^{0}_{2})$. Hence $\delta_{2}(E)=\delta_{2}(E^{0})+\mid Y\mid$.

Since $A\subseteq E$, we get that $Y$ is linearly independent over $\langle E^{0}_{2}C_{2}\rangle^{lin}\supseteq A_{2}$. Therefore this linear independence is given by the linear indpendence of $\{[x,y]:x\in X_{1}(C),y\in X_{1}(A)\}$ as in Corollary 3.9 described. Hence $\mathbb{F}(E^{*})=\mathbb{F}((E^{0})^{*})\otimes\mathbb{F}(\langle Y\rangle^{lin}$. By Corollary 3.9 again $E^{-}=\mathbb{F}(E^{*})/N$ implies that $N\subseteq\mathbb{F}((E^{0})^{*})$. Hence $\delta(E^{0})\leq\delta(E)$. ∎

In the following we identify the isomorphic copies of $B$ in $A$ and $C$ and denote them by $B$.

Theorem 6.15**.**

Assume that $B\leq A$ and $B\leq^{2(ldim(A/B))+2+n}C$ are in $\mathcal{K}_{3}^{fin}$. Then an amalgam $D$ of $A$ and $C$ over $B$ exists in $\mathcal{K}_{3}^{fin}$ with $C\leq D$ and $A\leq^{n}D$ . If $B\leq C$, then $A\leq D$. If no divisor problem of $B$ has a solution in both $A$ and $C$, then $D=A\otimes_{B}C$ has the desired properties. In this case $\delta(D)=\delta(A)+\delta(C)-\delta(B)$.

Proof.

Let $[b,?]=e$ be divisor problem in $B$ with solutions $a\in A$ and $c\in C$. Since $B\leq A$ we have $\langle Ba\rangle^{A}\cong B[e:b]=B^{\prime}$ and by $B\leq^{2(ldim(A/B))+2+n}C$ also $\langle Bc\rangle^{C}\cong B[e:b]=B^{\prime}$ . We use $B^{\prime}$ for $\langle Ba\rangle^{A}$ as a substructure of $A$ and also $B^{\prime}$ for $\langle Bc\rangle^{C}$ as a substructure of $C$. By Lemma 3.15 $\delta(B^{\prime})=\delta(B)$ and therefore $B^{\prime}\leq A$ and $B^{\prime}\leq^{2(ldim(A/B^{\prime})+2+n}C$. Hence it is sufficient to prove the assertion for $A,B^{\prime},C$. Using an appropriate induction we can assume that there is no divisor problem in $B$ with solutions in both $A$ and $C$. In this case we show that $D=A\otimes_{B}C$ has the desired properties. By Theorem 3.4 $D=A\otimes_{B}C$ exists. By Theorem 4.2 we know that $D^{*}\in\mathcal{K}_{2}^{fin}$.

Lemma 6.4 (4) implies $C\leq D$. Similarly we have $A\leq D$, if $B\leq C$.

Next we show $A\leq^{n}D$. We consider $A\subseteq E\subseteq D$ with $o-dim(E^{*}/A^{*})\leq n$. By Mon $\langle(C\cap E)A\rangle=(C\cap E)\otimes_{B}A$. By Lemma 6.14 we have $\delta((E\cap C)\otimes_{B}A)\leq\delta(E)$. Since $B\leq^{2(ldim(A/B))+2+n}C$ it holds $B\leq(E\cap C)$. Lemma 6.4(4) implies $A\leq(E\cap C)\otimes_{B}A$. Hence $\delta(A)\leq\delta((E\cap C)\otimes_{B}A)\leq\delta(E)$.

By Theorem 4.2 we have $D^{*}\in\mathcal{K}_{2}$. To show that $D\in\mathcal{K}_{3}^{fin}$ we consider $E\subseteq D$. We have to show: If $0<o-dim(E)$, then $\delta(E)\geq min\{2,o-dim(E)\}$. By Corollary 5.7 we have to consider only $\delta$. Since $D^{*}\in\mathcal{K}_{2}$ and by Corollary 5.7 we can assume w.l.o.g. that $E\leq_{2}D$. By Lemma 6.9 we have $(E\cap C)\leq E$. Therefore $\delta(E\cap C)\leq\delta(E)$.

Furthermore w.l.o.g. $o-dim_{3}(E)=0$. Since $C\in\mathcal{K}_{3}$ $0<\delta(E)$ for $(E\cap C)\not=\langle 0\rangle$ and $1<\delta(E)$ for $1<o-dim(E\cap C)$.

Now asume that $o-dim(E\cap C)\leq 1$. Then $o-dim(E\cap B)\leq 1$. An generating o-system for $E\cap A$ has the form $X(A)$ or $b,X(A)$, where

$b$ is a non-zero element of $B_{1}\cap E_{1}$ or $B_{2}\cap E_{2}$ and

$X(A)=X(A)_{1}X(A)_{2}$, where

$X(A)_{1}$ consists of elements $a_{i}^{1}$ with $a_{i}^{1}\in(A_{1}\cap E_{1})$ linerarly independent over $B_{1}$

and $X(A)_{2}$ consists of elements $a_{i}^{2}\in(A_{2}\cap E_{2})$ linearly independent over $\langle BX(A)_{1}\rangle_{2}$.

To get an o-system for the vector space $E\cap\langle CA\rangle^{lin}$ we have to extend the o-system above in the following way:

Either we add $c\in C_{1}\setminus B_{1}$ or $c\in C_{2}\setminus B_{2}$, if $E\cap C\not=\langle 0\rangle$ and $E\cap B=\langle 0\rangle$.

Furthermore we have $Y=Y_{1}Y_{2}$, where $Y_{j}$ consists of $d_{i}^{j}+e_{i}^{j}$, where the $d_{i}^{j}\in A_{j}$ are linearly independent over $\langle B_{j},\ldots a_{i}^{j}\ldots\rangle^{lin}$ and $e_{i}^{j}\in C_{j}$ linearly independent over $\langle B_{j},c\rangle^{lin}$, (sometimes there is no $c$).

Let $X$ be $X(A),b$, if there is some $b$ and $X=X(A)$ otherwise. By Corollary 3.9 we can extend $X,Y$ or $X,c,Y$ respectively by some Z to get a generating o-system for $E$, where $Z$ is linearly independent $\langle CA\rangle^{lin}$.

Let $\Phi=\Phi_{2}\Phi_{3}$ be an generating ideal system such that $E=F(XYZ)/\Phi$ or

$E=F(X,c,Y,Z)/\Phi$.

Since $o-dim(C\cap E)\leq 1$ we have $\Phi\subseteq F(X)$. Hence $\delta_{2}(E)=\delta_{2}(E\cap A)+\mid c,Y,Z\mid$ or $\delta_{2}(E)=\delta_{2}(E\cap A)+\mid Y,Z\mid$ respectively, and

$\delta(E)=\delta(E\cap A)+\mid c,Y,Z\mid$ or $\delta(E)=\delta(E\cap A)+\mid c,Y,Z\mid$ respectively. Then $min\{2,o-dim(E)\}\leq\delta(E)$.

∎

7. The strong Fraïssé Hrushovski Limit

$\mathcal{K}^{fin}_{3}$ is countable and there are only countably many strong embeddings for Lie algebras in $\mathcal{K}^{fin}_{3}$. By Theorem 6.15 we have the amalgamation property for strong embeddings. As well known we get the following strong Fraïssé Hrushovski limit. For more details see [22].

Theorem 7.1**.**

7.1 There exists a countable structure $\mathfrak{M}$ in $\mathcal{K}_{3}$ that satisfies the following condition:

• rich

If $B\leq A$ are in $\mathcal{K}_{3}^{fin}$ and there is a strong embedding $f$ of $B$ into $\mathfrak{M}$, then it is possible to extend $f$ to strong embedding of $A$ into $\mathfrak{M}$.

We call $\mathfrak{M}$ the strong Fraïssé Hrushovski limit of $\mathcal{K}_{3}^{fin}$.

Corollary 7.2**.**

$\mathfrak{M}$* is unique up to isomorphisms.*

The proof is similar to the proof of the next result.

Theorem 7.3**.**

For any two rich structures $M$ and $N$ in $\mathcal{K}_{3}$ with $\langle\bar{a}\rangle\leq M$, $\langle\bar{b}\rangle\leq N$, and $\langle\bar{a}\rangle^{M}\cong\langle\bar{b}\rangle^{N}$ we have $(M,\bar{a})\equiv_{L_{\infty,\omega}}(N,\bar{b})$.

Proof.

We will show that we can play the Fraïssé Ehrenfeucht game infinitely many rounds reproducing the starting condition $\langle\bar{a}\rangle\leq M$, $\langle\bar{b}\rangle\leq N$, and $\langle\bar{a}\rangle^{M}\cong\langle\bar{b}\rangle^{N}$ again and again. Assume w.l.o.g. that player I has chosen an element $c$ in $M$. Choose $\bar{c}$ in $M$ such that $\bar{c}$ contains $\bar{a}$ and $c$, and $\langle\bar{c}\rangle^{M}\leq M$. Then $\langle\bar{a}\rangle^{M}\leq\langle\bar{c}\rangle^{M}$. By assumption we have some isomorphism $f$ of $\langle\bar{a}\rangle^{M}$ onto $\langle\bar{b}\rangle^{N}\leq N$. By richness $f$ can be enlarged to embedding of $\langle\bar{c}\rangle^{M}$ onto some $\langle\bar{d}\rangle^{N}\leq N$. ∎

It follows that there is a common complete theory of all rich Lie algebras in $\mathcal{K}_{3}$. In Corollary 7.7 we will see that the following gives an axiomatization.

Definition 7.4**.**

1. $T_{3}(1)$

Let $T_{3}(1)$ be the elementary description of $\mathcal{K}_{3}$. 2. $T_{3}(2)$

For every pair $B\leq A$ in $\mathcal{K}_{3}$ we fix a function $g(n)$ (see next lemma), such that:

If $B\leq^{g(n)}M$, then this embedding of $B$ in $M$ can be extendent to a $\leq^{n}$-strong embedding of $A$ in $M$. 3. $(T_{3})$

We define $T_{3}=T_{3}(1)\cup T_{3}(2)$.

Theorem 6.15 implies the following

Lemma 7.5**.**

Every rich structure $M$ in $\mathcal{K}_{3}$ fulfills the elementary axioms $T_{3}(2)$.

Proof.

Let $C$ be the selfsufficient closure of $B$ in $M$. Then we have $B\subseteq C\leq M$. If $B\leq M$, then $B=C$. We assume that $g(n)=2(ldim(A/B))+2+n^{2}+n$ and $B\leq^{2(ldim(A/B))+2+n^{2}+n}C$. We apply Theorem 6.15 to get an amalgam $D$ in $\mathcal{K}_{3}$ of $A$ and $C$ over $B$, such that $C\leq D$ and $A\leq^{n^{2}+n}D$. Since $M$ is rich and $C\leq M$ there is a strong embedding of $D$ in $M$ over $C$. By Lemma 6.13 we get $A\leq^{n}M$.

∎

Theorem 7.6**.**

A Lie algebra $M$ in $\mathcal{K}_{3}$ is rich iff it is a $\omega$-saturated model of $T_{3}$.

Proof.

By Lemma 7.5 a rich $M\in\mathcal{K}_{3}$ satisfies $T_{3}$. A $\omega$-saturated model of $T_{3}$ is a rich Lie algebra in $\mathcal{K}_{3}$ by $\omega$-saturation. It remains to show that rich $M\in\mathcal{K}_{3}$ are $\omega$-saturated. Let $N$ be a $\omega$-saturated model of $T_{3}$. Then $N$ is rich. and by Theorem 7.3 we have $M\equiv_{L_{\infty,\omega}}N$. Hence $M$ is $\omega$-saturated. ∎

Corollary 7.7**.**

$T_{3}$* is an axiomatization of the complete theory of all rich graded Lie algebras in $\mathcal{K}_{3}$. If $M$ is a model of $T_{3}$ and $A$ is a strong substructure of $M$, then the elementary type of $A$ is completely determined by its isomorphism-type.*

Lemma 7.8**.**

A model $M$ of $T_{3}$ satisfies the following:

1. (1)

For all $1\leq i<j\leq 3$: $\forall xz\exists y(U_{i}(x)\wedge U_{j}(z)\rightarrow[x,y]=z)$. 2. (2)

$M^{*}\models T_{2}$** 3. (3)

If $M$ is rich, then $M^{*}$ is rich.

Proof.

To prove (1) assume $U_{i}(a)$ and $U_{j}(c)$. Let $B\leq M$ with $a,c\in B$. If $[a,?]=c$ has no solution in $B$, then $B\leq B(c:a)$ is in $\mathcal{K}_{3}$ by Lemma 3.15. By $T_{3}(2)$ there is a solution in $M$. For (2) and (3) it is sufficient to show that $M^{*}$ is rich. This follows since $M$ and therefore $M^{*}$ are $\omega$-saturated. ∎

Lemma 7.9**.**

Let $A$ and $B$ be subalgebras of some $M\in\mathcal{K}_{3}$, where $B=\langle B_{1},B_{2},b_{1}\ldots b_{k}\rangle$ and $b_{1}\ldots b_{k}\in B_{3}$ are linearly independent over $\langle B_{1}B_{2}\rangle_{3}$.

1. (1)

If $B\leq^{n+k}A$, then $\langle B_{1}B_{2}\rangle\leq^{n}A$. 2. (2)

If $B\leq A$, then $\langle B_{1}B_{2}\rangle\leq A$.

Proof.

(1) Choose $\langle B_{1}B_{2}\rangle\subseteq E\subseteq A$ with $o-dim(E^{*}/\langle B_{1}B_{2}\rangle^{*})\leq n$. Assume w.l.o.g. $B\cap E=\langle B_{1}B_{2}b_{1}\ldots b_{i}\rangle$. By assumption

[TABLE]

Hence $\delta(\langle B_{1}B_{2}\rangle)\leq\delta(E)$.

(2) followa from (1). ∎

8. Non-forking

Definition 8.1**.**

Let $X$ be any subalgebra of $M\models T_{3}$. We say $X$ is strong in $M$ (short: $X\leq M$), if for all finte $A\subseteq X$ we have $CSS(A)\subseteq X$. We define $CSS(Y)=\cup_{(A\subseteq Y,Afinite)}CSS(A)$ for $Y\subseteq M$.

Note that this definition generalizes the definition for finite $X$. Furthermore $CSS(Y)\leq M$. Remember that $A,B,C,D$ are only used for finite substructures. By Corollary 5.7 $B\leq M$ can be defined using $\delta$ only in the following way:

For all $A$ with $B\subseteq A\subseteq M$ we have $\delta(B)\leq\delta(A)$.

Lemma 8.2**.**

Let $M$ be a model of $T_{3}$.

1. (1)

$X\leq M$* if and only if for all finite $B\leq M$ $\delta(B\cap X)\leq\delta(B)$.* 2. (2)

If $X\leq M$ and $A\leq M$, then $X\cap A\leq M$.

Proof.

ad (1): First assume $X\leq M$. Consider $B\leq M$. By assumption $CSS(B\cap X)\subseteq X$. Hence $CSS(B\cap X)=B\cap X$. It follows $\delta(B\cap X)\leq\delta(B)$.

To prove the other direction consider $A\subseteq X$. Since $CSS(A)\leq M$ we get $\delta(CSS(A)\cap X)\leq\delta(CSS(A))$ by assumption. Since $A\subseteq(CSS(A)\cap X)$ we get $CSS(A)=CSS(A)\cap X\subseteq X$.

ad (2): Since $X\leq M$, there is some $D\leq M$ such that $A\cap X\subseteq D\subseteq X$. Then $A\cap X=A\cap D\leq M$. ∎

Let $\mathfrak{C}$ be a monster model of $T_{3}$ and $M\preceq\mathfrak{C}$. If $A\subseteq M$, then $CSS(A)$ belongs to the algebraic closure of $A$. Hence $CSS(A)\subseteq M$ and $M\leq\mathfrak{C}$. If $\bar{a}\in\mathfrak{C}$, then by $M\leq\mathfrak{C}$ and definition

[TABLE]

Note that $\delta(CSS(B\bar{a}))-\delta((CSS(B\bar{a})\cap M)\geq 0$.

Lemma 8.3**.**

*Assume $X\leq\mathfrak{C}$ and $\bar{a}\in\mathfrak{C}$. Choose $\bar{a}\subseteq A\leq CSS(\langle X\bar{a}\rangle)$, such that $\delta(A)-\delta(A\cap X)$ is minimal.

Then $CSS(\langle X\bar{a}\rangle)=X\otimes_{A\cap X}A$ and $A=CSS(\langle(A\cap X)\bar{a}\rangle)$.*

Proof.

Note $A\leq\mathfrak{C}$. If $(A\cap X)\leq C\leq X$, then by Lemma 6.3

[TABLE]

Note that $A\cap C=A\cap X$. Since $C\leq CSS(\langle AC\rangle)\cap X\leq X$, we have

• (*)

$\delta(CSS(\langle AC\rangle))/(CSS(\langle AC\rangle)\cap X)\leq\delta(CSS(\langle AC\rangle))/C)\\ \leq\delta(\langle AC\rangle/C)\leq\delta(A/A\cap C)$.

By minimality of $\delta(A/A\cap X)$ we have equality in () for all such $C$. Therefore $\delta(CSS(\langle AC\rangle))=\delta(\langle AC\rangle)$. Hence $CSS(AC)=\langle AC\rangle$ and $CSS(AX)=\langle AX\rangle$. By Lemma 6.8 we get $\langle AC\rangle=C\otimes_{A\cap C}A$, since $\delta(A/C)=\delta(A/A\cap C)$ by (). We get $CSS(AX)=\langle AX\rangle=X\otimes_{A\cap X}A$

Since $CSS(\langle(A\cap X)\bar{a}\rangle)\subseteq A$ and $CSS(\langle(A\cap X)\bar{a}\rangle)\cap X=A\cap X$, we have for $A^{\prime}=CSS(\langle(A\cap X)\bar{a}\rangle)$ that

[TABLE]

by the considerations above. Since $\delta(A)=\delta(A^{\prime})$, this implies $A=CSS(\langle(A\cap X)\bar{a}\rangle)$.

∎

Corollary 8.4**.**

$T_{3}$* is $\omega$-stable.*

Proof.

Let $M$ be any countable model of $T_{3}$ and $M\preceq\mathfrak{C}$. By Lemma 8.3 for $a\in\mathfrak{C}$ there are $B\leq M$ and $A$ such that $B=(M\cap A)\leq\mathfrak{C}$, $A=CSS(Ba)$, and $CSS(Ma)=M\otimes_{B}A$. $tp(a/M)$ is uniquely determined by the isomorphism type of $CSS(Ma)$ by Theorem 7.3. Hence it is sufficient to count for all $B\leq M$ all pairs $B\leq A$ in $\mathcal{K}_{3}^{fin}$ with $A=CSS(Ba)$ for some $a$. ∎

Lemma 8.5**.**

Let $C$ be a subalgebra of $\mathfrak{C}$. The algebraic closure of $C$ in $\mathfrak{C}$ ($acl(C)$) is the transitive closure of $CSS(C)$ under adjunction of homogeneous divisors.

Proof.

$CSS(C)$ is in the algbraic closure of $C$. A homogeneous divisor is in the algebraic closure. Let $X$ be the transitive closure of $CSS(C)$ under homogeneous divisors. Then $X$ is in the algebraic closure of $C$ and $X\leq\mathfrak{C}$ by Lemma 3.15.

Assume that $a$ is not in $X$. Choose $A$ as in Lemma 8.3, such that

$A=CSS(\langle(A\cap X)a\rangle)$ and $CSS(\langle Xa\rangle)=X\otimes_{X\cap A}A$. Let $X\cap A\leq A^{1}$ be isomorphic to $A$ over $X\cap A$. All homogeneous divisors for $X\cap A$ in $A^{1}$ are in $X\cap A$. Hence $(X\otimes_{X\cap A}A)\otimes_{X\cap A}A^{1}$ is in $\mathcal{K}_{3}$ by Theorem 6.15 applied to all $X\cap A\leq D\leq X$. Since $\mathfrak{C}$ is saturated we find $A^{1}$ in $\mathfrak{C}$. Then $(X\otimes_{X\cap A}A)\otimes_{X\cap A}A^{1}\leq\mathfrak{C}$. Since we can iterate this argument we see that $A$ and $a$ are not in the algebraic closure of $C$. ∎

Above we have already defined:

For subsets $A,B,C$ in a structure $M$ we define

[TABLE]

if and only if

[TABLE]

Since $T_{3}$ is $\omega$-stable, we have non-forking A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C.

Lemma 8.6**.**

Let $B\subseteq A$ and $B\subseteq C$ be substructures of a monster model $\mathfrak{C}$ of $T_{3}$. Then the following are equivalent:

1. (1)

A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C. 2. (2)

There are $A^{\prime},B^{\prime},C^{\prime}$, such that

1. (a)

$CSS(A)\subseteq A^{\prime}\subseteq acl(A)$, 2. (b)

$CSS(C)\subseteq C^{\prime}\subseteq acl(C)$, 3. (c)

$B^{\prime}=A^{\prime}\cap C^{\prime}\subseteq acl(B)$, 4. (d)

$\langle A^{\prime}C^{\prime}\rangle\cong A^{\prime}\otimes_{B^{\prime}}C^{\prime}$, and $\langle A^{\prime}C^{\prime}\rangle\leq\mathfrak{C}$.

Proof.

First we assume that (2) is true. Using Lemmas 3.15, 3.16, and 3.17 we can extend $B^{\prime}$ and $C^{\prime}$, such that every solution of a division problem of $B^{\prime}$ in $A^{\prime}$ is already in $B^{\prime}$. If we denote the new structures again by $A^{\prime},B^{\prime},C^{\prime}$, then they are again strong in $\mathfrak{C}$ and (2) remains true.

Let $C_{0}=C^{\prime},C_{1},\ldots$ be any insiscernible sequence over $B^{\prime}$. There are automorphisms $f_{i}$ of $\mathfrak{C}$ that fix $B^{\prime}$ pointwise with $f_{i}(C^{\prime})=C_{i}$. Let $E_{i}$ be $CSS(\langle C_{0}\ldots C_{i}\rangle)$. There are no new homogeneous divisors in $A^{\prime}$ for problems in $B^{\prime}$. By Theorem 6.15 $A^{\prime}\otimes_{B^{\prime}}E_{i}$ exists in $\mathcal{K}_{3}$. Since $\mathfrak{C}$ is rich, there is some $A_{i}\subseteq\mathfrak{C}$ such that

$\langle A^{\prime}E_{i}\rangle\cong\langle A_{i}E_{i}\rangle\cong A_{i}\otimes_{B^{\prime}}E_{i}$ and $\langle A_{i}E_{i}\rangle\leq\mathfrak{C}$. By saturation there exists $A_{\omega}$ such that $\langle A_{\omega}E_{i}\rangle\leq\mathfrak{C}$ and $\langle A^{\prime}E_{i}\rangle\cong\langle A_{\omega}E_{i}\rangle\cong A_{\omega}\otimes_{B^{\prime}}E_{i}$ for all $i<\omega$.

By Theorem 6.15 $A_{\omega}\leq A_{\omega}\otimes_{B^{\prime}}E_{i}$. $C_{i}\leq\mathfrak{C}$, since $C^{\prime}\leq\mathfrak{C}$. We have $CSS(\langle A_{\omega}C_{i}\rangle)\subseteq A_{\omega}\otimes_{B^{\prime}}E_{i}\leq\mathfrak{C}$. By the structure of a free product we get $\langle A_{\omega}C_{i}\rangle\leq A_{\omega}\otimes_{B^{\prime}}E_{i}\leq\mathfrak{C}$:

To show this consider $\langle A_{\omega}C_{i}\rangle\subseteq D\subseteq A_{\omega}\otimes_{B^{\prime}}E_{i}$. Then

[TABLE]

[TABLE]

By Lemma 6.14 we get

[TABLE]

Now we can applay Theorem 7.3 and get $tp(A_{\omega}/C_{i})=tp(A^{\prime}/C^{\prime})$ for all $i<\omega$. Then A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B^{\prime}}C^{\prime} and therefore A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}C, since $A\subseteq A^{\prime}\subseteq acl(A)$, $C\subseteq C^{\prime}\subseteq acl(C)$, and $B\subseteq B^{\prime}\subseteq acl(B)$.

Assume (1) holds. We extend $A^{\prime},B^{\prime}$ in (2 a, b, c) in such a way, that every homogeneous solution in $C^{\prime}$ of a divisor problem of $B^{\prime}$ is already in $B^{\prime}$. Use Lemmas 3.17, 3.16, and 3.15 . We use the same notation. Then $A^{\prime}\leq\mathfrak{C}$ and $C^{\prime}\leq\mathfrak{C}$ By assumption A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B^{\prime}}C^{\prime} and $B^{\prime}\subseteq acl(B)$. Define $C_{0}=C^{\prime}$. By Theorem 6.15 $C_{0}\otimes_{B^{\prime}}C_{1}$ exists in $\mathcal{K}_{3}$, where $C_{0}\cong C_{1}$ and both are strong in $C_{0}\otimes_{B^{\prime}}C_{1}$. Since $\mathfrak{C}$ is rich there exits $C_{1}$ in $\mathfrak{C}$, such that $C_{0}\otimes_{B^{\prime}}C_{1}\leq\mathfrak{C}$. Then $C_{1}\leq\mathfrak{C}$ and there is an automorphism$f_{1}$ of $\mathfrak{C}$ with $f_{1}(C_{0})=C_{1}$ and $f_{1}(C_{1})=C_{0}$. By induction we get a sequence $C_{0},C_{1},\ldots$ in $\mathfrak{C}$, such that $C_{i}\leq\mathfrak{C}$ and $E_{i}=C_{0}\otimes_{B^{\prime}}C_{1}\otimes_{B^{\prime}}\ldots\otimes_{B^{\prime}}C_{i}\leq\mathfrak{C}$. Every substructure generated by a finite subset of the $C_{i}$ is strong in $\mathfrak{C}$ and isomorphic to the free amalgam of these $C_{i}$ over $B^{\prime}$. (Use Theorem 6.15 and Theorem 7.3). Hence the sequence of the $C_{i}$ is an indiscernible sequence over $B^{\prime}$. Let $f_{i}$ be the automorphism of $\mathfrak{C}$, that exchanges $C_{0}$ and $C_{i}$ and fixes all the other $C_{j}$. Since A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B^{\prime}}C^{\prime} there is some $A_{\omega}$ in $\mathfrak{C}$, such that $tp(A_{\omega}/C^{\prime})=tp(A^{\prime}/C^{\prime})$ and $C_{0},C_{1}\ldots$ is indiscernible over $A_{\omega}$. Note that $B^{\prime}\subseteq A_{\omega}$. $f_{i}(tp(A_{\omega}/C^{\prime})=tp(A_{\omega}/C_{i})$. The following claim implies the assertion:

[TABLE]

(+) implies

[TABLE]

Then we have $\delta(\langle A^{\prime}C^{\prime}\rangle)=\delta(A^{\prime})+\delta(C^{\prime})-\delta(B^{\prime})$. The $\delta$’s of the old and the new $A^{\prime},B^{\prime},C^{\prime}$ are the same. Hence we have (++) also for the original $A^{\prime},B^{\prime},C^{\prime}$.

First we assume

[TABLE]

Then Lemma 6.8 implies

[TABLE]

Next we show, that (*) implies

[TABLE]

Otherwise

[TABLE]

and

[TABLE]

would imply by Sym and Trans

[TABLE]

and by Mon a contradiction to (*). Then

[TABLE]

By induction we get:

[TABLE]

since otherwise equality would be equivalent to A_{\omega}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{\langle C_{0}\ldots C_{i-1}\rangle}C_{i} and this implies together with C_{i}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{B^{\prime}}\langle C_{0}\ldots C_{i-1}\rangle

[TABLE]

in contradiction to (*) by Mon.

Hence there is some j such that

[TABLE]

This contradicts

[TABLE]

We have shown the first part of the claim.

Using the same steps we can show, that

[TABLE]

Then

[TABLE]

It remains to prove

[TABLE]

Otherwise we have

[TABLE]

We show by induction on i that this implies

[TABLE]

Then

[TABLE]

for sufficiently large j - a contradiction. We denote

[TABLE]

by $D_{i}$. Then $\delta(A_{\omega})\leq\delta(D_{i})$ , since $A_{\omega}\leq\mathfrak{C}$. For the induction we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

Corollary 8.7**.**

1. (1)

Assume $B\leq A\leq\mathfrak{C}$ and $B\leq X\leq\mathfrak{C}$, such that $A\cap X=B$ where $X$ can be infinite. Then A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}X if and only if A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\otimes}_{B}X and $\langle AX\rangle\leq\mathfrak{C}$ if and only if $\delta A/X)=\delta(A/B)$ and $\langle AX\rangle\leq\mathfrak{C}$ . 2. (2)

Types over strong substructures are stationary.

We come back to Lemma 8.3. Note that every element in $\mathfrak{C}$ is interdefinable with a sequence of homogeneous elements.

Lemma 8.8**.**

Assume $U\leq V\leq\mathfrak{C}$, $\bar{a}\subseteq\mathfrak{C}$, $B\leq U$, $CSS(B\bar{a})=A_{U}$, $A_{U}\cap U=B$, and

[TABLE]

Assume $acl(\langle U\bar{a}\rangle)\cap V=U$, $C\leq V$, , $A_{V}=CSS(C\bar{a})$, $A_{V}\cap V=C$, and

[TABLE]

If $D=B\cap C$ and $A=A_{U}\cap A_{V}$, then $A=CSS(\langle D\bar{a}\rangle)$ and

[TABLE]

Proof.

Extend $\bar{a}$ to $\bar{e}$, such that $\langle\bar{e}\rangle\cap U=\langle\bar{a}\rangle\cap U=\langle\bar{a}\rangle\cap V$ and $\bar{e}$ generates $A$ over $D$.

By Mon $\langle U\bar{e}\rangle=U\otimes_{B}\langle B\bar{e}\rangle$ and $\langle V\bar{e}\rangle=V\otimes_{C}\langle C\bar{e}\rangle$. Then Theorem 3.10 implies

[TABLE]

We get $\langle BA\rangle=\langle B\bar{e}\rangle=B\otimes_{D}\langle D\bar{e}\rangle$.

If $\langle BA\rangle=A_{U}$, then $\langle U\bar{e}\rangle=CSS(\langle U\bar{a}\rangle)$, as desired.

Otherwise there is some $c\in A_{U}$, that is not in $\langle BA\rangle$. Then $c$ is not in $U$, not in $V$, and not in $A_{V}$.

If we consider $c$ as a linear combination over a vector space basis of

[TABLE]

as described in Lemma 3.9, then there are mixed monomials with elements in $V\setminus C$ and $A_{V}\setminus C$. But then $c$ is not involved in any relations. Therefore $c\notin CSS(\langle B\bar{a}\rangle=A_{U}$, a contradiction. ∎

In the next Corollary we use Lemma 8.8 for the case $U=V$.

Corollary 8.9**.**

*Assume $U\leq\mathfrak{C}$ and $\bar{a}\in\mathfrak{C}$. According to Lemma 8.3we have: If $A$ is chosen with $\bar{a}\subseteq A\leq CSS(\langle U\bar{a}\rangle)$, such that $\delta(A)-\delta(A\cap U)$ is minimal,

then $CSS(\langle U\bar{a}\rangle)=U\otimes_{A\cap U}A$ and $A=CSS(\langle(A\cap U)\bar{a}\rangle)$.

There is an $A$ as above, that is minimal with these properties. We call $B=A\cap U$ the self-sufficient canonical base algebra of $tp(\bar{a}/U)$.*

Lemma 8.10**.**

We consider $U\leq\mathfrak{C}$ and $tp(\bar{a}/U)$. Let $B$ be the self-sufficient base algebra of $tp(\bar{a}/U)$, as in Corollary 8.9. Then $B$ is a weak canonical base of $tp(\bar{a}/U)$. With respet to the given properties it is unique.

Proof.

We claim that $B$ is a weak canonical base of $tp(\bar{a}/U)$. By Corollary 8.7 we have U\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}A. Since $\mathfrak{C}$ is saturated, we can replace $U$ by a saturated model $U\leq M\preceq\mathfrak{C}$ such that M\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{B}A. We have the same situation for $M$ as for $U$:

$A=CSS(\langle B\bar{a}\rangle)=\langle B\bar{e}\rangle$, $A\cap M=B$, $CSS(M\bar{a})=M\otimes_{B}A$. and $A$ minimal with this properties.

Now we use Lemma 8.8 for $M=N$. Let $f$ be an automorphism of $\mathfrak{C}$ that fixes $M$ setwise.

First we assume, that it fixes $B$ pointwise. If $g$ is $f$ for $M$ and the identity for $\bar{a}$, then it can be extended to $CSS(M\bar{a})=M\otimes_{B}A$. Since $M\otimes_{B}A\leq\mathfrak{C}$, $g$ is an elementary automorphism and therefore $tp(\bar{a}/M)=f(tp(\bar{a}/M)$ .

Conversely assume that $f$ fixes $tp(\bar{a}/M)$. W.l.o.g. we can assume that $f$ fixes $\bar{a}$ pointwise. Then $f(A)\cap M=f(B)$. By Lemma 8.8 we get $CSS(M\bar{a})=M\otimes_{B\cap f(B)}\langle(B\cap f(B))\bar{e}\rangle$. Hence $B=f(B)$. ∎

Corollary 8.11**.**

$T_{3}$* has the weak elimination of imaginaries.*

In section 10 we will introduce minimal strong prealgebraic extensions over strong subalgebras. We have canonical base algebras for them. We use $CB(\bar{a}/X)$ to denote the canonical base of $tp(\bar{a}/X)$.

Let us come back to the Lemma 8.9.Now we assume that $\bar{a}$ is generating o-system over $B$ and over $X$.

[TABLE]

Let $X^{B}$ be a generating o-system of $B$. Then $B^{*}=F(X^{B})^{*}/\Phi_{2}^{B}$, where $F(X^{B})$ is the free algebra freely generated by $X^{B}$ and $\Phi_{2}^{B}\subseteq F(X^{B}_{1})^{*}_{2}$. Furthermore $B^{-}=\mathbb{F}(B^{*})/\Phi_{3}^{B}$, where $\Phi_{3}^{B}\subseteq\mathbb{F}(B^{*})_{3}$. Again we use elements of $\langle B\bar{a}\rangle$ to denote their preimages in $F(X^{B}\bar{a})$ and in $\mathbb{F}(\langle B\bar{a}\rangle^{*})$.

The extension of $X$ above is completely determined by $\langle B\bar{a}\rangle$. There are sets $\Phi_{2}$ and $\Phi_{3}$ such that

[TABLE]

[TABLE]

where $\Phi_{2}\subseteq F(X^{B}\bar{a})^{*}_{2}$ is linearly independent over $F(X^{B})$ and $\Phi_{3}\subseteq\mathbb{F}(\langle B\bar{a}\rangle^{*})$ is linearly independent over $\mathbb{F}(B^{*})$.

See Definitions 2.6 and 2.7.

Definition 8.12**.**

The elements of $\Phi_{i}$ for $i=2,3$ are called relations of degree i.

Their structure is described in the proof of Theorem 3.10. We have coefficients and ends.

Corollary 8.13**.**

In Lemma 8.9 $B$ is the self-sufficient closure of the substructure of $X$ generated by the ends and the coefficients of an ideal basis $\Phi_{2}\Phi_{3}$ in the situation above.

Corollary 8.14**.**

$T_{3}$* is CM-trivial.*

Proof.

Assume $M\preceq N\preceq\mathfrak{C}$ and $acl(\bar{a}M)\cap N=M$. We have to show that $CB(\bar{a}/M)\subseteq acl^{eq}(CB(\bar{a}/N))$.

Let $B$ be the self-sufficient canonical base algebra of $tp(\bar{a}/M)$ and $C$ be the self-sufficient canonical base algebra of $tp(\bar{a}/N)$. We have to show, that $B\subseteq C$.

Then we have $B\leq M$, $CSS(B\bar{a})=A_{M}$, $A_{M}\cap M=B$, and

[TABLE]

Furthermore $acl(\langle M\bar{a}\rangle)\cap N=M$, $C\leq N$, , $A_{N}=CSS(C\bar{a})$, $A_{N}\cap N=C$, and

[TABLE]

By Lemma 8.8

[TABLE]

where $A=CSS((B\cap C)\bar{a})$. Since $B$ is minimal we get $B\subseteq C$, as desired.

∎

9. geometry

Let $\mathfrak{C}$ be a monster model of $T_{3}$ again. Let $U,V,W$ be substructures of $\mathfrak{C}$ in $\mathcal{H}_{3}$. They are generated by their projections into $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ . If they are finite we use often $H,J,K$. $\mathcal{H}_{3}^{fin}$ is the subset of the finite substructures in $\mathcal{H}_{3}$. Since substructures are graded, the set of all $H\in\mathcal{H}_{3}$ of o-dimension $\leq 1$ is $\{\langle a\rangle:a\in(\mathfrak{C}_{1}\cup\mathfrak{C}_{2})\}$. Note that for $a_{1}\in\mathfrak{C}_{1}$ and $a_{2}\in\mathfrak{C}_{2}$ we have $\langle a_{1}+a_{2}\rangle=\langle a_{1},a_{2}\rangle$ and o-dimension and $\delta$ of this substructure is 2.

To compute $\delta(H)$ for $H\in\mathcal{H}_{3}^{fin}$ , we consider $H\cong\mathbb{F}(H^{*})/N$. Then $\delta(H)=\delta_{2}(H)-ldim(N)$.

We define a pregeometry on $R(\mathfrak{C})=\mathfrak{C}_{1}\cup\mathfrak{C}_{2}$. For this we define a dimension function for all $H\in\mathcal{H}_{3}^{fin}$.

Definition 9.1**.**

The dimension of a finite subset $X=X_{1}\cup X_{2}$ with $X_{1}\subseteq\mathfrak{C}_{1}$ and $X_{2}\subseteq\mathfrak{C}_{2}$ of $R(\mathfrak{C})$ is the dimension of the substrucure $H=\langle X\rangle\in\mathcal{H}_{3}^{fin}$ generated by these elements: $d(H)=\delta(CSS(H))$.

$d(A/H)=d(\langle AH\rangle)-d(H)$ and $d(A/U)=min\{d(A/H):H\subseteq U\}$.

We have $CSS(H)=\langle CSS(H)_{1}CSS(H)_{2}\rangle$. Furthermore $d(\langle 0\rangle)=0$

Lemma 9.2**.**

$d$* defines a dimension function on $R(\mathfrak{C})$.*

Proof.

For $H$ and $K$ in $\mathcal{H}_{3}$ we have to show:

1. (1)

$d(\emptyset)=d(\langle 0\rangle)=0$. 2. (2)

$d(H)>0$, if $H\not=\langle 0\rangle$, since $\mathfrak{C}\in\mathcal{K}_{3}$. 3. (3)

$d(\langle Ha\rangle)\leq d(H)+1$ for $a\in R(\mathfrak{C})$. 4. (4)

If $K\subseteq H$, then $d(K)\leq d(H)$. 5. (5)

$d(\langle HK\rangle)\leq d(H)+d(K)-d(\langle(H_{1}\cap K_{1})(H_{2}\cap K_{2})\rangle)$.

• ad(3)

If $a\in CSS(H)$, then $d(H)=d(\langle Ha\rangle)$. Otherwise we use $CSS(H)\leq\langle CSS(H)a\rangle$. Then $\mathbb{F}(CSS(H)^{*})\subseteq\mathbb{F}(\langle CSS(H)a\rangle^{*})$. Hence

[TABLE]

• ad(4)

By Lemma 6.11 we get $CSS(K)\subseteq CSS(H)$. Hence $d(K)\leq d(H)$.

• ad(5)

$CSS(H)$ and $CSS(K)$ are strong subsructures of $CSS(\langle HK\rangle)$. By Theorem 6.3

[TABLE]

[TABLE]

$CSS(\langle(H_{1}\cap K_{1}),(H_{2}\cap K_{2})\rangle)\leq CSS(H)\cap CSS(K)$. Hence $d(\langle(H_{1}\cap K_{1})(H_{2}\cap K_{2})\rangle)=\delta(CSS(\langle(H_{1}\cap K_{1})(H_{2}\cap K_{2})\rangle))\leq\delta(CSS(H)\cap CSS(K))$

∎

We define $cl$ for substructures $H$ and $U$ of $\mathfrak{C}$ in $\mathcal{H}_{3}$ and on the set $R(\mathfrak{C})$.

Definition 9.3**.**

For $H\subseteq\mathfrak{C}$ and $H\in\mathcal{H}_{3}^{fin}$ we define

[TABLE]

$cl(U)$ is the union of all $cl(H)$ for $H\subseteq U$.

For $X\subseteq R(\mathfrak{C})$ we define

[TABLE]

Then the following is well-known:

Corollary 9.4**.**

$cl$* is a pregeometry on $R(\mathfrak{C})$. Furthermore $acl(H)\subseteq cl(H)$. If $H\leq K\leq\mathfrak{C}$, then $K\subseteq cl(H)$ if and only if $\delta(K)=\delta(H)$.*

Lemma 9.5**.**

Assume $U\leq\mathfrak{C}$, $a\not\in U$, and $a\in R(\mathfrak{C})$. Then there are the following possibilities:

1. (1)

$d(a/U)=1$. In this case $\langle Ua\rangle=CSS(Ua)$. (Transcentental Case) 2. (2)

$d(a/U)=0$.

1. (a)

$CSS(Ua)=\langle Ua\rangle$* . Then there are $u_{1}\in U_{i}(U)$ and $u_{2}\in U_{j}(U)$ such that $i<j$ and $\mathfrak{C}\models[u_{1},a]=u_{2}$. (Algebraic Case)* 2. (b)

$\delta(a/U)>0$.

Definition 9.6**.**

We consider $U\in\mathcal{H}_{3}$, $U\leq\mathfrak{C}$ and $V\in\mathcal{H}_{3}$ finitely generated over $U$.

• •

If $V=\langle Ua\rangle$ for some $a\in R(\mathfrak{C})$ and $a\not\in cl(U)$, then we call $V$ a minimal trancentental extension.

• •

If $V=\langle Ua\rangle$ for some $a\in R(\mathfrak{C})$ and $a\in acl(U)$, then we call $V$ a minimal algebraic extension.

• •

If $V\not=\langle Ua\rangle$ for all $a\in R(\mathfrak{C})$, $\delta(V/U)=0$, and $\delta(W/U)>0$ for all $U\subseteq W\subseteq V$, $U\not=W\not=V$, and $W\in\mathcal{H}_{3}$, then $V$ is minimal prealgebraic over $U$. (short prealgebraic)

• •

We use the definition of minimal prealgebraic also for $U$ not strong in $\mathfrak{C}$.

If $H\leq K$ and $K$ is minimal strong, then $K$ is generated over $H$ by a single element or it is prealgebraic minimal (Lemma 9.5.

Lemma 9.7**.**

Assume $U\leq V\leq\mathfrak{C}$ and $V/U$ is finite. Then there is a sequence $U=V_{0}\leq V_{1}\leq\ldots\leq V_{n}=V$, such that $V_{i+1}$ is minimal strong over $V_{i}$.

10. Prealgebraic minimal extensions

Lemma 10.1**.**

Assume that $H\leq K\leq\mathfrak{C}$ is prealgebraic minimal.

1. (1)

If $H\leq U\leq\mathfrak{C}$, then $K\leq U$ or $U\cap K=H$ and $\langle UK\rangle$ is a minimal prealgebraic extension of $U$ and $\langle UK\rangle=U\otimes_{H}K$. $tp(K/U)$ is the unique nonforking extension of $tp(K/H)$. 2. (2)

$tp(K/H)$* is strongly minimal.*

Proof.

By Lemma 6.3 we have

[TABLE]

If $H\not=K\cap U\not=K$, then $\delta(K/(K\cap U))<0$, a contradiction. If $K\cap U=H$, then

[TABLE]

and therefore $\langle UK\rangle=U\otimes_{H}K$ by Lemma 6.8.

∎

Now we consider a prealgebraic minimal strong extension $H\leq K\leq\mathfrak{C}$. W.l.o.g we can assume that there is some $M\preceq\mathfrak{C}$, such that $tp(K/M)$ with $K\cap M=H$ is a nonforking extension. By Lemma 10.1:

[TABLE]

By Lemma 8.9 we get $A\leq\langle MK\rangle$ and $B=M\cap A$, such that

[TABLE]

$o-dim_{3}(A/B)=0$, $A=\langle B\bar{a}\rangle$ where $\bar{a}$ is a generating o-system of $A$ over $B$, and $B$ is a self-sufficient weak canonical base algebra of $tp(A/M)$.

$B$ and $A$ are strong substructures and minimal with these properties. $B$ is in $\mathcal{K}_{3}$. Now we will only assume that $A$ is $\leq^{m}$-strong for sufficiently large $m$.

Lemma 10.2**.**

Assume $U\leq\mathfrak{C}$, $\bar{a}$ is a generating o-system of $\langle U\bar{a}\rangle\leq\mathfrak{C}$ over $U$, $o-dim_{3}(\bar{a})=0$, and $\langle U\bar{a}\rangle$ is prealgebraic minimal extension of $U$.

1. (1)

If we have $\langle U\bar{a}\rangle=U\otimes_{B}\langle B\bar{a}\rangle$ and $\langle U\bar{a}\rangle=U\otimes_{C}\langle C\bar{a}\rangle$.

Then $\langle U\bar{a}\rangle=U\otimes_{B\cap C}\langle(B\cap C)\bar{(}a)\rangle$. 2. (2)

If we assume that $B\ \leq^{k}\mathfrak{C}$ and $C\leq^{k}\mathfrak{C}$ for $o-dim(B^{*}),o-dim(C^{*})<k$, then $(B\cap C)\leq^{k}\mathfrak{C}$.

For $D\subseteq B$ we have $D\leq^{k}\mathfrak{C}$ if and only if $D\leq B$.

Proof.

(1) is is a consequnece of Corollary 3.10.

(2) follows from Corollary 6.9.

∎

Definition 10.3**.**

For every prealgebraic minimal extension $B\leq A=\langle B\bar{a}\rangle\leq\mathfrak{C}$, where $B$ is the self-sufficient canonical base algebra as in Lemma 8.9 and $\bar{a}$ is a generating o-system of $A$ over $B$ we define a formula $\phi(\bar{x},\bar{y})$. $\bar{x}$ stands for $\bar{a}$ and $\bar{y}$ for a generating o-system for $B$. $\phi$ describes the following:

1. (1)

The isomorphism-type of $A$ over $B$. 2. (2)

There is no proper subalgebra $D\subseteq B$, such that $\langle D\bar{a}\rangle\leq A$ and $\langle D\bar{a}\rangle$ describes a minimal prealgebraic extension of $D$. 3. (3)

For $A/B$ we fix some $m(A/B)=m$, such that $2\;lindim(A/B)+2<m$ and $h(\mid\bar{x}\mid)+\mid\bar{x}\mid<m$ where $h$ is the function in Lemma 4.7. Then the formula says, that $B\leq^{m+o-dim(A^{*}/B^{*})}\mathfrak{C}$.

Let $\mathcal{X}^{home}$ be the set of these formalas. Instead of $\phi(\bar{a},\bar{b})$ we often write $\phi(A/B)$.

In the above definition $\phi(A/B)$ implies , that $A$ is a prealgebraic minimal strong extension of $B$, $A$ is $\leq^{m}$-strong in $\mathfrak{C}$, $B$ is minimal with this properties.

Lemma 10.4**.**

Let $B\leq A$ be in $\mathfrak{C}$ with $\mathfrak{C}\models\phi(A/B)$ for $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$, as described above. Assume $B\subseteq V$ for $V=\langle V_{1}V_{2}\rangle$, $A\not\subseteq V$, and $\delta(A/V)\geq 0$. Then

[TABLE]

$A$* provides a prealgebraic minimal strong extension of $V$.*

Proof.

We have $\langle VA\rangle\in\mathcal{H}_{3}$. We can asssume w.l.o.g. that $V$ is finite. Otherwise we work with sufficiently large substructures. Let $V^{s}$ be the selfsufficient closure of $V$ in $\langle VA\rangle$. By definiton of the selfsufficient closure we have $\delta(A/V^{s})\geq 0$ and

$\langle VA\rangle\not=V^{s}$, since $\delta(\langle VA\rangle)\geq\delta(V)$.

Now $V^{s}$ fullfils all assumptions on $V$ in the Lemma. We show the Lemma for $V^{s}$. Then it follows $V=V^{s}$ and the assertion for $V$. Hence we can assume w.l.o.g. that $V$ is strong in $\langle VA\rangle$.

By Corollary 5.7 $0\leq\delta_{2}(A/V)$. For $\mathfrak{C}^{*}$ Lemma 4.7 implies that there are $X_{V}\subseteq\langle V_{1}A_{1}\rangle_{2}\cap V_{2}$ linearly independent over $\langle V_{1}\rangle_{2}+\langle A_{1}\rangle_{2}+(V\cap A)_{2}$, $D$, and $H$, such that $A^{*}\cap V^{*}\subseteq H\subseteq D\subseteq V^{*}$ , $D_{1}=V_{1}$, $V^{*}=\langle DX_{V}\rangle$, $o-dim(H/(A^{*}\cap V^{*}))\leq h(o-dim(A^{*}/V^{*}))$ and

• (i)

$\langle VA\rangle^{*}=D\otimes_{H}\langle HA^{*}\rangle$.

Let $E_{1}\subseteq V_{1}$ and $E_{2}\subseteq V_{2}$ in $V$ be the isomorphic preimages of $H_{1}$ and $H_{2}$. Then define $E=\langle E_{1}E_{2}A\rangle\cap V$. Then $E^{*}$ is the given $H$ and $\langle E\cap V\rangle=E$. Furthermore we get $o-dim(E^{*}/A^{*})\leq o-dim(E^{*}/A^{*}\cap E^{*})\leq h(o-dim(A^{*}/V^{*}))\leq m$, where $m$ comes from the definition of $\phi$. This we need later for (v).

Since $V\leq_{2}\langle AV\rangle$ we get

• (ii)

$E\leq_{2}\langle EA\rangle$

by Lemma 4.1.

By Lemma 6.4(1) the conditions (i) , (ii) and $V\cap\langle EA\rangle=E$ imply

• (iii)

$0\leq\delta(A/V)\leq\delta(A/E),$

(iv) follows from Lemma 6.4(3), and (i) and (ii) since $V\leq\langle VA\rangle$.

• (iv)

$E\leq\langle EA\rangle$.

As seen before (ii) we have $o-dim(E^{*}/A^{*})\leq m$. Hence we have $A\leq\langle AE\rangle$ by $\mathfrak{C}\models\phi(A/B)$.

• (v)

$E$ and $A$ are strong in $\langle EA\rangle$.

By Lemma 6.3 and (iii)

• (vi)

$0\leq\delta(A/V)\leq\delta(A/E)\leq\delta(A/E\cap A).$

Hence

• (vii)

$E\cap A=B$,

since otherwise the definition of a minimal prealgebraic extension and (vi) would imply

$0\leq\delta(A/V)\leq\delta(A/E)\leq\delta(A/E\cap A)<0,$

a contradiction. Then (vi) and (vii) imply

• (viii)

$0\leq\delta(A/V)\leq\delta(A/E)\leq\delta(A/B)=0$

It follows by Corollary 6.8:

• (ix)

$\langle EA\rangle=E\otimes_{B}A$

By Lemma 6.4(2) we get

• (x)

If $0<\mid X_{V}\mid$, then $\delta(A/V)<\delta(A/E).$

By (x) and (viii) it follows that $X_{V}$ is empty. Hence we can apply Lemma 6.4(5) and get:

• (xi)

$\langle VA\rangle=V\otimes_{E}A$.

By Trans (ix) and (xi) imply

• (xii)

$\langle VA\rangle=V\otimes_{B}A$.

∎

Lemma 10.5**.**

Assume $\phi\in\mathcal{X}^{home}$, and $\mathfrak{C}\models\phi(A/B)$. Furthermore $B\subseteq V\leq\mathfrak{C}$ and $V\in\mathcal{H}_{3}$. Then

1. (1)

Either $A\subseteq V$ or $\langle AV\rangle=V\otimes_{B}A$ and $\langle AV\rangle\leq\mathfrak{C}$ is a minimal prealgebraic extension of $V$. 2. (2)

In both cases $A\subseteq cl(V)$.

Proof.

The assertions (1) and (2) are direct consequences of Lemma 10.4. A minimal prealgebraic extension of a strong substructure is strong.

∎

Corollary 10.6**.**

For $B\subseteq\mathfrak{C}$ and $\phi\in\mathcal{X}^{home}$ with $\mathfrak{C}\models\exists\bar{x}\phi(\bar{x},B)$ we have $\phi(\bar{x},B)$ is strongly minimal.

Proof.

Assume $\mathfrak{C}\models\phi(A/B)$. The pair $B\leq A$ is in $\mathcal{K}_{3}$ and there is no $x\in A\setminus B$ with $[e,x]=d$, where $e,d\in B$, since the extension is prealgebraic minimal strong. Let $B\subseteq C\leq M\prec\mathfrak{C}$. By $\mathfrak{C}\models\phi(A/B)$ $B$ is sufficiently strong in $C$ for the application of Theorem 6.15. Hence $C\otimes_{B}A^{\prime}$ exist as a substructure of $\mathfrak{C}$, where $tp(A^{\prime}/B)=tp(A/B)$. By saturation this remains true, if we replace $C$ by a model $M$:

[TABLE]

Hence there is a realisation of $\phi(x,B)$ generic over $M$. By Corollary 10.5 every solution $A$ that is not in $M$ has this form:

[TABLE]

Hence $\phi(\bar{x},B)$ is strongly minimal. ∎

Lemma 10.7**.**

Let $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$ and $\mathfrak{C}\models\exists\bar{x}\phi(\bar{x}/B)$. Then $B$ is the canonical paramter up to some automorphisms of $B$.

Proof.

Let $M\preceq\mathfrak{C}$ be saturated, such that $B\subseteq M$. By Lemma 10.6 there is a solution $\bar{a}$ of $\phi(\bar{x},B)$ generic over $M$. We use $A=\langle B\bar{a}\rangle$. Then $A\cap M=B$

Let $f$ be an automorphism of $\mathfrak{C}$ that fixes $B$ pointweise and $M$ setwise. Then $tp(\bar{a}/M)=tp(f(\bar{a})/M)$ by Lemma 10.5 and Corollary 10.6.

Now we consider an automorphism $f$ of $\mathfrak{C}$, such that $f$ fixes $M$ setwise and $f(tp(\bar{a}/M))=tp(\bar{a}/M)$. W.l.o.g. $f$ fixes $\bar{a}$ pointwise. By Lemma 10.5

[TABLE]

We assume $B\not=f(B)$ and show a contradiction. By Lemma 10.2 we get

[TABLE]

and $\langle(B\cap f(B))\bar{a}\rangle$ is again $\leq^{m}$-strong in $\mathfrak{C}$ and it is a minimal prealgebraic extension over a smaller substructure. See Lemma 6.4.

∎

Definition 10.8**.**

Assume $\bar{b}\subseteq V$ for $V\in\mathcal{H}_{3}$, $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$, and $\bar{a}$ is a solution of $\phi(\bar{x},\bar{b})$. Then $\bar{a}$ is weakly generic over $V$, if $\langle\bar{a}\bar{b}\rangle\cap V=\langle\bar{b}\rangle$ and $\delta(\bar{a}/V)=0$. (short \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic).

Lemma 10.4 implies:

Corollary 10.9**.**

Assume $\bar{b}\subseteq V$ for $V\in\mathcal{H}_{3}$, $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$, and $\bar{a}$ is a weakly generic solution of $\phi(\bar{x},\bar{b})$ over $V$. Then $\langle V\bar{a}\rangle=V\otimes_{\bar{b}}\langle\bar{b}\bar{a}\rangle$. $\bar{a}$ is a prealgebraic minimal strong extension of $V$.

By Lemma 10.7 we obtain

Corollary 10.10**.**

Let $\phi(\bar{x},\bar{y})$ be a formula in $\mathcal{X}^{home}$. If $\phi(\bar{x},\bar{b})$ and $\phi(\bar{x},\bar{b^{\prime}})$ have common generic solutions, then $\langle\bar{b^{\prime}}\rangle=\langle\bar{b}\rangle$ and there is an isomorphism of these two Lie - algebras, that can be extended by the identity of the common generic extension.

Hence we can define:

Definition 10.11**.**

For every $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$ there is a unique $\psi(\bar{x},y)\in L^{eq}$, such that for every $\bar{b}$ with $\mathfrak{C}\models\exists\bar{x}\phi(\bar{x},\bar{b})$ there is some $b\in\mathfrak{C}^{eq}$, such that $\phi(\bar{x},\bar{b})$ and $\psi(\bar{x},b)$ have the same solutions and $b$ is the canonical parameter of $\psi(\bar{x},y)$. Let $\mathcal{X}$ be the set of these $\psi$.

11. $T_{3}$ satisfies the conditions for the collapse

In [6] there are conditions P(I) - P(VII) formulated for a theory $T$ that provide the existence of an infinite substructure of the monster model of $T$ with an $\aleph_{1}$-categorical theory, such that $cl$ becomes the algebraic closure in the small structure. In that paper $R(\mathfrak{C})$ is a vector space and we work essentially with finite subspaces of it. Here $R(\mathfrak{C})$ is $\mathfrak{C}_{1}\cup\mathfrak{C}_{2}$ as defined above. We consider pairs of subspaces $V_{1}\subseteq\mathfrak{C}_{1}$ and $V_{2}\subseteq\mathfrak{C}_{2}$ . Often we work with the $L$ - structures $\langle V_{1}V_{2}\rangle$ in $\mathcal{H}_{3}$. The clousure $cl$ can be considered as a relation beween such $L$ -structures in $\mathcal{H}_{3}$. Elements coresspond to structures of o-dim 1. That means they correspond to the elements of $\mathfrak{C}_{1}\cup\mathfrak{C}_{2}$.

Now we transform the properties P(I) - P(VII) from [6] into conditions C(I) - C(VII) and show that they are true in models of our theory $T_{3}$. Then we can follow the proofs in [6]. Some modifications are necessary. We obtain the desired Lie algebra, that is a counterexample to Zil’ber’s conjecture.

Let $M$ be a model of $T_{3}$. We work with $R(M)$ , the pregeometry $cl$ on $R(M)$, the set $\mathcal{X}$ of formulas, and \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w} - genericity, as we have defined in this paper. Let $\mathfrak{C}$ be again the monster model of $T_{3}$.

1. C(1)

The models $M$ of $T_{3}$ are graded 3-nilpotent Lie algebras over $\mathbb{F}_{q}$, where $\mathbb{F}_{q}$ is a finite field. The graduation is given by $U_{1},U_{2},U_{3}$.

We work with $R(x)=U_{1}(x)\cup U_{2}(x)$ in $M$. Then $M=\langle R(M)\rangle$.

C(1) is clear

1. C(2)

There is a pregeometry on $R(M)$, given by $a\in cl(H)$ for $a\in R(M)$ and $H\in\mathcal{H}_{3}$ and a notion ” $H$ is a strong substructure of $M$”, short $H\leq M$. Both notions are invariant under automorphisms of $\mathfrak{C}$. $\langle 0\rangle\leq M$. For every finite $H$ there exists an finite algebraic extension that is strong in $M$. Algebraic extensions of strong subspaces in $\mathcal{H}_{3}$ are strong. If $M$ and $N$ are models of $T_{3}$, $H\subseteq R(M)$, $K\subseteq R(N)$, $tp^{M}(H)=tp^{N}(K)$, and $a\in M_{i}$ and $b\in N_{i}$ for $i=1$ or $i=2$ are geometrically independent of $H$ and $K$ respectively, then $tp^{M}(a,H)=tp^{N}(b,K)$. If in this case $H\leq M$, then $\langle Ha\rangle\leq M$. The geometrical dimension $d(\mathfrak{C})$ of $R(\mathfrak{C})$ is infinite.

C(2) is proved in this paper, especially in section 9. The next condition is proved in section 10. $\mathcal{X}$ is defined there.

1. C(3)

There is a set ${\mathcal{X}}$ of formulas $\psi(\bar{x},y)$ in $L^{eq}$ such that $\psi(\bar{x},b)$ is either empty or strongly minimal.

If $\psi(\bar{a},b)$ is true, then $b$ is the canonical parameter. It is the canonical base algebra $B$ modulo some automorphisms. $\bar{a}=\bar{a}_{1}\bar{a}_{2}$ with $\bar{a}_{1}\subseteq U_{1}$ and $\bar{a}_{2}\subseteq U_{2}$ defines an o-system over $B$. If $\psi(\bar{x},b)$ and $\psi(\bar{x},b^{\prime})$ have common generic solutions, then $b=b^{\prime}$.

Length$(\bar{x})=n_{\psi}\geq 2$. If $b$ is in ${\rm dcl}^{eq}(U)$ and $M\models\psi(\bar{a},b)$, then $\bar{a}\in cl(U)$. If furthermore $U\leq M$, then either $\bar{a}\subseteq U$ or $\bar{a}$ is a generic solution over $U$. In the generic case $\langle U\bar{a}\rangle\leq M$. ${\mathcal{X}}$ is closed under affine o-transformations. See below.

We want to explane, what it means that ${\mathcal{X}}$ is closed under affine o-transformations:

We consider $\psi(\bar{x},y)$ in $\mathcal{X}$. Let $\phi(\bar{x},\bar{y})$ be a corresponding formula in $\mathcal{X}^{home}$. By Definition 10.3 and Lemma 10.4 we have that $\mathfrak{C}\models\phi(\bar{a},\bar{b})$, $\bar{a}\notin U$, $\bar{b}\in U,$ and $\delta(\bar{a}/U)\geq 0$ implies

[TABLE]

If $\bar{c}\in\langle U\bar{a}\rangle\setminus U$ is an o-system over $U$ with

[TABLE]

then $\bar{c_{1}}=H_{1}(\bar{a_{1}})+\bar{m_{1}}$, where $H_{1}$ is a vector space automorphism of $\langle\bar{a_{1}}\rangle$ and $\bar{m_{1}}$ is a tuple in $U_{1}$.

Furthermore $\bar{c_{2}}=H_{2}(\bar{a_{2}})+H_{d}(\bar{a_{1}})+\bar{m_{2}}$, where $H_{2}$ is a vector space automorphism of $\langle\bar{a_{2}}\rangle$, $H_{d}(\bar{a_{1}})=(\ldots,\sum_{j}[d^{i,j},a_{1}^{j}],\ldots)$, where $d^{i,j}$ is in $U_{1}$, and $\bar{m_{2}}$ is a tuple in $U_{2}$. The matrix $H_{d}$ uses the Lie-multiplication.

The tuple $H_{1}(\bar{a_{1}}),H_{2}(\bar{a_{2}})+H_{d}(\bar{a_{1}})$ we denote by $H(\bar{a})$.

Definition 11.1**.**

We call $H(\bar{a})$ an o-transformation over $B$, if all $d^{i,j}\in B$. Furthermore $H(\bar{a})+\bar{m_{1}m_{2}}$ is an affine o-transformation over $B$.

$H(\bar{a})+\bar{m_{1}m_{2}}$ gives us a new generating o-system of $\langle U\bar{a}\rangle$ over $U$. As above we get a a formula in $\mathcal{X}^{home}$ for $\phi(H(\bar{x})+\bar{m},\bar{b})$. An o-transformation $H(\bar{x})$ is an homomorphism of the the underlying vector space of $\mathfrak{C}^{n_{\alpha}}$.

It is easily seen:

Lemma 11.2**.**

Assume that $U\leq\mathfrak{C}$, $\langle U\bar{a}\rangle=\langle U\bar{e}\rangle$ are a minimal prealgebraic extension of $U$ given by $\mathfrak{C}\models\phi(\bar{a},\bar{b})$ and $\mathfrak{C}\models\psi(\bar{e},\bar{c})$ with $\phi,\psi\in\mathcal{X}^{home}$ and $\bar{b}$ and $\bar{c}$ are in $B\subseteq U$. Then the generating o-system $\bar{e}$ of $\langle U\bar{a}\rangle$ over $U$ is given by an affine o-transformation of $\bar{a}$ over $B$.

Proof.

By the considerations above we have $\bar{e}=H(\bar{a})+\bar{m}$, where $H$ is an o-transformation and $\bar{m}\in U$. By Lemma 10.4

[TABLE]

and

[TABLE]

Since every element of $\bar{e}_{2}$ is involved in some relations, these relations have to split. We get that $H$ is an o-transformation over $B$. ∎

Also the next condition we have proved:

1. C(4)

Let $H\leq M$, $K\leq M$. If $\langle H\rangle\cong\langle K\rangle$, then $tp(H)=tp(K)$. If $H\leq K\leq M$, $H\subseteq K\subseteq cl(H)$, then there is a chain $H=K_{0}\leq K_{1}\leq\ldots\leq K_{n}$, where $K_{n}=K$, $K_{i}\leq M$, and $K_{i+1}\subseteq acl(K_{i})$ or $K_{i+1}$ is obtained from $K_{i}$ by adding a solution of some $\psi(\bar{x},b)$ in $\mathcal{X}$ generic over $K_{i}$, where $b\subseteq dcl^{eq}(K_{i})$.

1. C(5a)

Let $\psi(\bar{x},y)\in\mathcal{X}$, $V$ a subspace in $\mathcal{H}_{3}$, and $b\in dcl^{eq}(V)$. Then the \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}-generic type of $\psi(\bar{x},b)$ over $V$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $V$ and all \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generics of $\psi(\bar{x},b)$ over $V$ have the same isomorphism type over $V$. They are \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over every $U\subseteq V$ with $b\in dcl^{eq}(U)$. 2. C(5b)

If $\psi(\bar{x},y)\in\mathcal{X}$, $U\leq M$, $b\in dcl^{eq}(H)$, and $\bar{e}_{0},\bar{e}_{1},\ldots$ are solutions of $\psi(\bar{x},b)$ with $\bar{e}_{i}\not\subseteq\langle U,H,\bar{e}_{0},\ldots,\bar{e}_{i-1})\rangle$, then there are at most $\delta(H/U)$ many $i$ such that $e_{i}$ is not \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle U,H,\bar{e}_{0},\ldots,\bar{e}_{i-1}\rangle$.

The first statements of C(5) follow from 10.4 and 10.9. To prove the last assertion, we consider

[TABLE]

where $\delta^{j}=\delta(\bar{e_{j}}/\langle U,H,\bar{e_{0}},\ldots,\bar{e_{j-1}}\rangle)$.

If $\delta^{j}\geq 0$, then $e_{j}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle U,H,\bar{e}_{0},\ldots,\bar{e}_{j-1}\rangle$ by Lemma 10.4. This proves the assertion.

Definition 11.3**.**

Let $X$ be a definable subset of $\mathfrak{C}^{n}$ of Morley - degree 1. We consider $\mathfrak{C}^{n}$ as a group with respect to $+$.

$X$ is called a group set (respectively torsor set) if its generic type is the generic type of a definable subgroup of $\mathfrak{C}^{n}$ (respectively coset of a definable subgroup). $X$ is groupless if it is not a torsor set.

Let $D$ be a subalgebra of $\mathfrak{C}$. Furthermore $\phi(\bar{x},\bar{e})$ is in $\mathcal{X}^{home}$ with $\bar{e}\in D$ and there are at least 2 solutions in $D$, that form an o-system over $\bar{e}$. We assume, that $D^{*}\cong F(X)/I$, where $X$ is a generating o-system of $D^{*}$ that contains the o-system $\bar{e}_{1}\bar{e}_{2}$ and $F(X)$ is the free 2-nilpotent graded Lie algebra over $X$. The isomorphism-type of the $U_{1}\oplus U_{2}$-part of the solution set of $\phi$ is given by linear combinations $\Theta$ of monomials $[x,y]$ where $x$ is from $\bar{x}_{1}$ and $y$ from $\bar{e}_{1}$ or $\bar{x}_{1}$ and of one element $g$ from $\langle\bar{e}\rangle_{2}$. We call $\Theta$ a relation of level 2.

Let $\bar{a}$ be a solution. Since $g$ is also a term over the second solution, we can assume w.l.o.g. that $\Theta(\bar{a},\bar{e})\in I$.

Then we consider $D^{-}=\mathbb{F}(D^{*})/J$. Similary as above we define relations $\Delta$ of level 3. Again it contains one summand $h$ from $\langle\bar{e}\rangle_{3}$ and we can consider it as a term over the second solution. Hence w.l.o.g. $\Delta(\bar{a},\bar{e})\in J$.

Corollary 11.4**.**

We are in the situation above. A solution of $\phi(\bar{x},\bar{e})$ in $D$ is given by relations of levels 2 and 3 that are in $I$ or $J$ respectively.

1. C(6)

Assume $C\supseteq B\subseteq A$ are strong subspaces of $\mathfrak{C}$ in $\mathcal{H}_{3}^{fin}$ with $C\cap A=B$ and both minimal strong extensions of $B$ given by generic solutions over $B$ of formulas in $\mathcal{X}$. Furthermore there are $e\in dcl^{eq}(E)$, $E\in\mathcal{H}_{3}$, $E\subseteq\langle AC\rangle$, and at least two solutions in $\langle CA\rangle$ of some $\psi(\bar{x},e)$ in ${\mathcal{X}}$. Let $\bar{d}\in\langle AC\rangle$ be a solution of $\psi(\bar{x},e)$ in ${\mathcal{X}}$ \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle CE\rangle$ and over $\langle AE\rangle$.

Then $\psi(\bar{x},e)$ defines a torsor set. If it defines a group set, then $e$ is in ${\rm dcl}^{\rm eq}(B)$.

We can assume, that $C$ and $A$ are defined by formulas in $\mathcal{X}^{home}$ with canonical base algebras in $B$.

We prove C(6). By assumption we have $\delta(C)=\delta(B)=\delta(A)$ and $B\leq\mathfrak{C}$ by Corollary 6.9. By Lemma 6.3 we get

[TABLE]

Since $B\leq\mathfrak{C}$ it follows $\delta(\langle CA\rangle)=\delta(B)$ and $\langle CA\rangle\leq\mathfrak{C}$. Since $\delta(A/C)=\delta(A/B)$, we have by Lemma 6.12

[TABLE]

$D$ is in $\mathcal{H}_{3}^{fin}$. Assume $(D^{*})=F(X)/I$ and $D^{-}=\mathbb{F}(D^{*})/N$, where $X$ is a generating o-system of $D^{*}$, $F(X)$ is the free 2-nilpotent Lie algebra over $X$. Later we will modify $X$. Then $I\subseteq F(C_{1})_{2}+F(A_{1})_{2}$. Since $C\leq\mathfrak{C}$ and $A\leq\mathfrak{C}$ we can suppose w.l.o.g. that $\mathbb{F}(C^{*})\subseteq\mathbb{F}(D^{*})$, $\mathbb{F}(A^{*})\subseteq\mathbb{F}(D^{*})$, and $\mathbb{F}(B^{*})=\mathbb{F}(C^{*})\cap\mathbb{F}(A^{*})\subseteq\mathbb{F}(D^{*})$.

We obtain $N\subseteq\mathbb{F}(C^{*})_{3}+\mathbb{F}(A^{*})_{3}$.

Let $\phi(\bar{x},\bar{e})\in\mathcal{X}^{home}$ be the corresponding formula for $\psi(\bar{x},e)$, , where $e$ is $\bar{e}$ modulo some automorphisms and $\langle\bar{e}\rangle\subseteq E$.

We can assume, that $\bar{e}=\bar{e_{1}}\bar{e_{2}}\bar{e_{3}}$ is a generating o-system for $\langle\bar{e}\rangle$.

We assume that the solution $\bar{d}$ of $\psi(\bar{x},y)$ consists of $d^{i}_{1}\in U_{1}$ for $1\leq i\leq m$ and $d^{i}_{2}\in U_{2}$ for $m<i\leq n$.

Let $X^{B}=X_{1}^{B}X_{2}^{B}X_{3}^{B}$ with $X_{i}^{B}\subseteq B_{i}$ be a vector space basis of $B$.

We extend $X^{B}$ by $X^{C}$ to get a vector space basis for $C$ and

by $X^{A}$ to obtain a vector space basis for $A$. $X^{C}$ and $X^{A}$ should be graded as $X^{B}$.

We can extend $X^{B}X^{C}X^{A}$ to obtain a vector space basis of $C\otimes_{B}A$ as in Corollary 3.9. Corollary 11.4 implies, that $\bar{d_{i}}\subseteq C_{i}+A_{i}$ for $i=1,2$, since these elements have to be involved in relations of $I$ or $N$.

The same is true for elements in $\bar{e}$ . Here elements in $C_{3}+A_{3}$ are possible.

Hence the elements $d_{1}^{j}\in\mathfrak{C}_{1}$ of $\bar{d}$ have to be in $C_{1}+A_{1}$ and the elements $d_{2}^{j}$ have to be in $C_{2}+A_{2}$. Since $\bar{d}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w} - generic over $\langle CE\rangle$ and $\langle AE\rangle$, we get $d_{1}^{j}=c_{1}^{j}+a_{1}^{j}$, where the $c_{1}^{j}\in C_{1}$ are linearly independent over $A_{1}+E_{1}$ and the $a_{1}^{j}\in A_{1}$ are linearly independent over $C_{1}+E_{1}$ and

$d_{2}^{j}=c_{2}^{j}+a_{2}^{j}$, where the $c_{2}^{j}\in C_{2}$ are linearly independent over $A_{2}+E_{2}$ and the $a_{2}^{j}\in A_{2}$ are linearly independent over $C_{2}+E_{2}$ .

Since $\bar{d}$ is a weakly generic solution over $\langle CE\rangle$ and $\langle AE\rangle$ Corollary 10.9 implies

[TABLE]

and

[TABLE]

Note that mixed monomials $[x,y]$ with $x\in X_{1}^{C}$ and $y\in X_{1}^{A}$ are linearly independent over $C_{2}+A_{2}$ and cannot be involved in any relation of $I$.

We choose the vector space bases above with the following additional assumptions for $i=1,2,3$:

$X^{B}$ contains a vector space bases $V_{i}^{B}$ with elements $e_{i}^{B,j}$ for $\langle\bar{e_{i}}\rangle\cap B$, where $i=1,2,3$.

$X^{C}$ contains a vector space bases $V_{i}^{C}$ with elemets $e_{i}^{C,j}$ for $\langle\bar{e_{i}}\rangle\cap C$ modulo $B_{i}$.

$X^{A}$ contains a vector space bases $V_{i}^{A}$ with elements $e_{i}^{A,j}$ for $\langle\bar{e_{i}}\rangle\cap A$ modulo $B_{i}$.

Furthermore there are $f_{i}^{j}\in X_{i}^{C}$ and $g_{i}^{j}\in X_{i}^{A}$, such that $V_{i}^{B}V_{i}^{C}V_{i}^{A}$ together with the basis $V_{i}^{+}$ of elements $f_{i}^{j}+g_{i}^{j}$ form a vector space basis of $\langle\bar{e}\rangle_{i}\cap(C_{i}+A_{i})$.

The $c_{i}^{j}$ and $a_{i}^{j}$ are linearly independent over $V_{i}^{B}V_{i}^{A}V_{i}^{C}\ldots f_{i}^{j}\ldots g_{i}^{j}\ldots$. We assume that they are part of our basis $X^{B}X^{C}X^{A}$. An element $\Theta$ in $I\cap\langle\bar{e_{1}}\bar{d_{1}}\rangle^{F(D_{1})}_{2}$ has the form:

[TABLE]

where $h_{1}^{j}\in\langle\bar{e_{1}}\rangle\cap B_{1}$ , $\Theta_{c}\in F(C_{1})_{2}$ , and $\Theta_{a}\in F(A_{1})_{2}$ , and $\Theta_{c}+\Theta_{a}\in\langle\bar{e}\rangle_{2}$. Here we use the description of a vector space basis of $C\otimes_{B}A$ and $I\subseteq\langle C_{1}\rangle_{2}^{F(D_{1})}+\langle A_{1}\rangle_{2}^{F(D_{1})}$. We call the $h_{1}^{j}$ the coefficients of $\Theta$.

Corollary 3.9 implies that the description of a vector space basis for the representation of $\Theta$ as above work also for $\langle CE\bar{d}\rangle=\langle CE\rangle\otimes_{\langle\bar{e}\rangle}\langle\bar{e}\bar{d}\rangle$ and $\langle AE\bar{d}\rangle=\langle AE\rangle\otimes_{\langle\bar{e}\rangle}\langle\bar{e}\bar{d}\rangle$. We see that the relations of level 2 define a torsor set. In the case of a group set there are no $\Theta_{c}+\Theta_{a}$ and we have only the coefficients in $B$.

Now we turn to level 3: $\mathbb{F}(D^{*})_{3}$. We need an order of a subspace of $D_{1}$. We define $V_{1}^{B}<V_{1}^{C}<V_{1}^{A}<V_{1}^{+}<c_{1}^{j}<a_{1}^{j}$. Inside $V_{1}^{B}$, $V_{1}^{C}$, $V_{1}^{A}$, and $V_{1}^{+}$ the order is given by the indizies. Furthermore $c_{1}^{j}<a_{1}^{j}<c_{1}^{k}<a_{1}^{k}$, if $j<k$. We consider

[TABLE]

$\langle\bar{e}\bar{d}\rangle_{2}^{\mathbb{F}(D^{*})}\cap(\mathbb{F}(C^{*})+\mathbb{F}(A^{*}))_{2}$ has the following vector space basis:

$V_{2}=V_{2}^{B}V_{2}^{C}V_{2}^{A}V_{2}^{+}V_{2}^{0}V_{2}^{1}$, where $V_{2}^{1}=\{d_{2}^{j}:m<j\leq n\}$ and $V_{2}^{0}$ is a subset of $\{[d_{1}^{j},e_{1}^{B,k}]:1\leq j\leq m,k\}$. By assumption the $[c_{1}^{j},e_{1}^{B,k}]$ as summands of $[d_{1}^{j},e_{1}^{B,k}]\in V_{2}^{0}$ are linearly independent over $A_{2}+E_{2}$. The same is true for the $[a_{1}^{j},e_{1}^{B,k}]$ over $C_{2}+E_{2}$.

In the next application of Corollary 3.9 we use $[[d_{1}^{j},d_{1}^{k}],x]=[[d_{1}^{j},x],d_{1}^{k}]-[[d_{1}^{k},x],d_{1}^{j}]$.

Then the following set $W$ is a vector space basis of $\langle\bar{e}\bar{d}\rangle_{3}^{\mathbb{F}(D^{*})}$ over $\mathbb{F}(C^{*})_{3}+\mathbb{F}(A^{*})_{3}$:

[TABLE]

where

1. (1)

$W_{1}=\{[d_{2}^{j},x]:m<j\leq n,x\in V_{1}^{C}\cup V_{1}^{A}\cup V_{1}^{+}\cup\{d_{1}^{k}:1\leq k\leq m\},\}$ 2. (2)

$W_{2}=\{[[d_{1}^{i},d_{1}^{j}],d_{1}^{k}]:j<i,j\leq k\}$, 3. (3)

$W_{3}=\{[y,d_{1}^{k}]:1\leq k\leq m,y\in V_{2}\setminus(V_{2}^{B}\cup V_{2}^{1})\},$ 4. (4)

$W_{4}=\{[y,d_{1}^{k}]:1\leq k\leq m,y=[x,z],((x\in V_{1}^{C}\wedge z\in V_{1}^{A}\cup V_{1}^{+})\vee(x\in V_{1}^{A}\wedge z\in V_{1}^{+})\vee(x=f_{1}^{i}+g_{1}^{i}\wedge z=f_{1}^{j}+g_{1}^{j}\wedge i<j),$ 5. (5)

$W_{5}=\{[[d_{1}^{i},z],v]:1\leq i\leq m,z<v,z,v\in V_{1}^{C}V_{1}^{A}V_{1}^{+}\}$. 6. (6)

$W_{6}=\{[[d_{1}^{i},z],d^{k}_{1}]:1\leq i\leq k\leq m,z\in V_{1}^{C}V_{1}^{A}V_{1}^{+}\}.$

Finally we consider elements $\Delta\in N\cap\langle\bar{e}\bar{d}\rangle_{3}^{\mathbb{F}(D^{*})}$. They need to have the following form:

[TABLE]

where $\Delta_{c}\in\mathbb{F}(C^{*})_{3}$, $\Delta_{a}\in\mathbb{F}(A^{*})_{3}$, $\Delta_{c}+\Delta_{a}\in\langle\bar{e}\rangle_{3}^{\mathbb{F}(D^{*})}$, and $\Delta_{d}$ is a sum of monomials over $\bar{e}\bar{d}$ that contain at least one $d_{i}^{j}$. By the linear independence of $W$ over $\mathbb{F}(C^{*})_{3}+\mathbb{F}(A^{*})_{3}$ we get that $\Delta_{d}$ is a linear combination of monomials $[d_{2}^{i},h_{1}]$, $[d^{i}_{1},h_{2}]$, and $[[d_{1}^{i},h_{1}^{j}],h_{1}^{k}]$, where the coefficients $h_{1}$, $h_{1}^{j}$, and $h_{1}^{k}$ are all in $B_{1}\cap\langle\bar{e}\rangle_{1}$ and $h_{2}$ is in $B_{2}\cap\langle\bar{e}\rangle_{2}$. By the form of the $\Theta$ and $\Delta$ we see that $\psi(\bar{x},e)$ defines a torsor set in $R(\mathfrak{C})^{n}$. If it is a group set, then we have only the coefficients in $B$.

1. C(7)

For every substructure $X$ of $M$ with $acl(R(X))\cap R(M)=R(X)$, $X_{1}\not=\langle 0\rangle$, and $\langle R(X)\rangle=X$, we have that $X$ is the set of all $d=a_{1}+a_{2}+[c_{1},c_{2}]$, where $a_{1},c_{1}\in R(X)_{1}$ and $a_{2},c_{2}\in R(X)_{2}$.

C(7) is clearly fulfilled. We have shown:

Theorem 11.5**.**

$T_{3}$* satiesfies the conditions C(1) - C(7).*

The proof of (C6) implies the following

Corollary 11.6**.**

*Assume $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$ corresponds to $\psi(\bar{x},y)\in\mathcal{X}$.

If $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$ gives us a torsor set, then isomorphism type of $\bar{x}$ over $\bar{y}$ is given by equations of the following form:*

[TABLE]

where $z^{j}\in\langle\bar{y}\rangle_{1}$ and $y^{2}\in\langle\bar{y}\rangle_{2}$ or

[TABLE]

*where the $w^{j}$ are monomials of the form $[x_{2}^{k},v_{1}]$ with $v_{1}\in\langle\bar{y}\rangle_{1}$ or of the form $[x_{1}^{k},v_{2}]$ with $v_{2}\in\langle\bar{y}\rangle_{2}$, or of the form $[[x_{1}^{k},v_{1}^{l}],v_{1}^{i}]$ with $v_{1}^{l},v_{1}^{i}\in\langle\bar{y}\rangle_{1}$ and $y^{3}\in\langle\bar{y}^{3}\rangle$.

A group set, given by a formula in $\mathcal{X}$, has no $y^{2}$ and $y^{3}$ in the equations.

If the set defined by $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$ is a torsor set, then $\phi(\bar{x},\bar{y})\in\mathcal{X}^{home}$ defines itself a coset of a group. In case of a group set it defines a group.*

Proof.

First we assume that $\phi$ gives a group set. In the setting of (C6) let $A$ and $C$ be extension of $B$, given by generic solutions $\bar{a}$ and $\bar{c}$ of $\phi(\bar{x},\bar{b})$ over $B$. By definition $\bar{a}+\bar{c}$ is a solution of $\phi(\bar{x},\bar{b})$ weakly generic over $A$ and $C$. Then the proof of (C6) gives the assertion. If $\phi$ gives a torsor set, then let $\bar{a}$ be again a generic solution of $\phi$ and $\bar{c}$ a solution of the corresponding group set. ∎

12. Codes and Difference Sequences

To get our main result we follow the proof in [6]. We replace the conditions P(I) - P(VII) in that paper by C(1) - C(7). $R(\mathfrak{C})$ is the union of two connected vector spaces now. According to this we have two kinds of elements in $R(\mathfrak{C})$. Therefore we consider substructures in $\mathcal{H}_{3}$ instead of the vector subspaces in [6].

Our approach in this paper is not strict axiomatic using C(1) - C(7), as in [6]. The next Lemma corresponds to Lemma 3.6 in [6].

Lemma 12.1**.**

Assume $M$ is a model of $T_{3}$, $A,U\in\mathcal{H}_{3}$ with $A\leq M$and $U\leq M$. Then

1. ( i)

$A\cap U\leq M$, and 2. (ii)

If $A\cap U\in\mathcal{H}_{3}$, $d(A/U)=d(A/A\cap U)$, then $\langle UA\rangle\leq M$. There is a sequence of minimal strong extensions for $A$

[TABLE]

For every such sequence for $A$ over $A\cap U$

[TABLE]

is a sequence of minimal strong extensions for $\langle UA\rangle$ over $U$.

Proof.

ad(i) For finite $U$ (i) is Lemma 6.9. Otherewise there is a finite $U_{0}\leq M$ with $A\cap U\subseteq U_{0}\subseteq U$. Then $A\cap U=A\cap U_{0}\leq M$.

ad(ii) By Lemma 9.7 we get a geometrical sequence $A\cap U=B_{0}\leq B_{1}\leq\ldots\leq B_{k}=A$. That means $B_{i+1}$ is minimal strong over $B_{i}$. All $B_{i}$ are strong in $M$. We show by induction on $i$ that $\langle UB_{i}\rangle\leq M$. We have $\langle UB_{0}\rangle=U\leq M$. If $B_{i+1}=\langle B_{i},b\rangle$ where $b$ is algebraic over $B_{i}$, then $\langle UB_{i}b\rangle\leq M$ by C(2). In the prealgebraic case the assertion follows from $\langle UB_{i}\rangle\leq M$ and C(3). In the transcendental case $B_{i+1}=\langle B_{i}b\rangle$ with $b\notin{\rm cl}(B_{i})$ we have $b\notin{\rm cl}(\langle B_{i}U\rangle)$ since $d(A/U)=d(A/B_{0})$. By C(2) $\langle B_{i+1}U\rangle\leq M$. ∎

Definition 12.2**.**

Two definable sets $X$ and $Y$ of Morley degree 1 are equivalent, if ${\rm MR}(X)={\rm MR}(Y)$ and ${\rm MR}(X\Delta Y)<{\rm MR}(X)$. We write $X\sim Y$.

Definition 12.3**.**

We consider definable sets $X\subseteq\mathfrak{C}^{n}$ where the elements $\bar{a}=\bar{a_{1}}\bar{a_{2}}$ are o-systems with $U_{1}(a_{1}^{j})$ and $U_{2}(a_{2}^{j})$. If $H(\bar{x})$ is an o-transformation, then we use $H(X)=\{H(\bar{a}):\bar{a}\in X\}$. $X+\bar{m}=\{\bar{a_{1}}+\bar{m_{1}},\bar{a_{2}}+\bar{m_{2}}:\bar{a}\in X\}$ for $\bar{m}=\bar{m_{1}}\bar{m_{2}}$ with $U_{1}(\bar{m_{1}})$ and $U_{2}(\bar{m_{2}})$.

The next 3 lemmas concern countable $\omega$-stable theories. In 12.5 2) we consider only $T_{3}$. For background and proofs see also [6]:

Lemma 12.4**.**

*(Version of Martin Ziegler [21])

Assume that $T$ is a countable $\omega$-stable theory of expansions of abelian groups. Let $M$ be a model of $T$ and $a$, $b$, $c$ be elements of $M$ with $a+b+c=0$ and pairwise independent over some set $B$. Then we have:*

The strong types of the elements $a$, $b$, $c$ over $B$ have the same stabilizer $U$ and $U$ is connected. 2. 2)

$a$, $b$, and $c$ are generic elements of ${\rm acl}(B)$-definable cosets of $U$. 3. 3)

It follows that $a$, $b$, and $c$ have the same Morley rank over $B$ namely ${\rm MR}(U)$. $U$ is definable over ${\rm acl}(B)$.

Lemma 12.5**.**

Let $X$, $Y$ be definable sets of Morley degree $1$.

If $X\sim Y$, $X,Y\subseteq\mathfrak{C}^{n}$, and $X$ is a group set $($resp. torsor set$)$, then $Y$ is a group set $($resp. torsor set$)$. 2. 2)

We work in $T_{3}$. We assume $X$ is defined by some $\phi(\bar{x},\bar{b})$ in $\mathcal{X}^{home}$ in $\mathfrak{C}^{n}$ and $H(X)$ is a o-transformation over $\bar{d}$. If $X$ is a group set then $H(X)$ is a group set. If $X$ is a torsor set then $H(X)+\bar{m}$ is a torsor set.

Proof.

ad(2) If $X$ is a group set, then $H(X)$ is a group set by Corollary 11.6. For $H(X)$ there is a formula in $\mathcal{X}^{home}$ over $\langle\bar{b}\bar{d}\rangle$. ∎

Lemma 12.6**.**

Let $\varphi(\bar{x},\bar{y})$ be a formula such that $\mathfrak{C}\vDash\exists\bar{x}\,\varphi(\bar{x},\bar{b})$ implies that $\varphi(\mathfrak{C},\bar{b})$ is a strongly minimal subset of $\mathfrak{C}^{n}$. Then $\{\bar{b}:\varphi(\mathfrak{C},\bar{b})\mbox{ is a group set}\}$ is definable. Similarly for torsor sets.

Proof. We consider the group case. The following statements are equivalent:

1. i)

$\varphi(\mathfrak{C},\bar{b})$ is a group set. 2. ii)

There exist two generic $\bar{b}$-independent realizations $\bar{a}_{1}$ and $\bar{a}_{2}$ of $\varphi(\bar{x},\bar{b})$ such that $\mathfrak{C}\vDash\varphi(\bar{a}_{1}+\bar{a}_{2},\bar{b})$. 3. iii)

$\mathfrak{C}\vDash\exists^{\infty}\,\bar{x}_{1}\in\mathfrak{C}^{n}\;\exists^{\infty}\,\bar{x}_{2}\in\mathfrak{C}^{n}(\varphi(\bar{x}_{1},\bar{b})\wedge\varphi(\bar{x}_{2},\bar{b})\wedge\varphi(\bar{x}_{1}+\bar{x}_{2},\bar{b}))$.

The equivalence of i) and ii) is shown in [10]. iii) is first order since $\varphi(\bar{x},\bar{b})$ is strongly minimal. It is clearly equivalent with ii). $\Box$

Note that $X$ is a torsor set if for some (every) $x\in X$ the set $X-x$ is a group set.

Definition 12.7**.**

Given a group set $X$ defined by a formula in $\mathcal{X}^{home}$ over $\bar{b}$, its invariant group is the set ${\rm Inv}(X)$ of o-transformations $H$ over $\bar{b}$ with $H(X)\sim X$.

For strongly minimal $\varphi(\bar{x},\bar{b})$   ”$H\in{\rm Inv}(\varphi(\bar{x},\bar{b}))$” is an elementary property of $\bar{b}$. As in [10] Lemma 12.4 implies

Lemma 12.8**.**

*Let $X\subseteq\mathfrak{C}^{n}$ be defined by a formula $\phi(\bar{x},\bar{b})\in\mathcal{X}^{home}$, where $\bar{b}\subseteq B$. $H$ is an o-transformation over $\bar{b}$.

Let $\bar{e}_{0}$ and $\bar{e}_{1}$ be two generic $B$-independent elements in $X$. If \bar{e}_{0}-H\bar{e}_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}\limits_{B}\bar{e}_{0} , then $X$ is a torsor set. Moreover, if $X$ is a group set, then $H$ is in ${\rm Inv}(X)$.*

Proof.

Note that $H(\bar{e_{j}})$ and $\bar{e_{j}}$ are interdefinable over $B$. We have

[TABLE]

Hence by Lemma 12.4 X is a torsor set. By Corollary 11.6 $\phi(\bar{x},\bar{b})$ defines the coset. In the group case $H(\bar{e_{1}})$ is a generic element of a coset of $X$. By Lemma 12.5 $H(X)$ is a group set, in fact a group. Hence $X=H(X)$. ∎

Lemma 12.9**.**

1. a)

Let $\varphi(\bar{x},y)$ and $\psi(\bar{x}_{0},\ldots,\bar{x}_{\mu},y)$ be formulas where $\bar{x}$ and $\bar{x}_{i}$ are in the home sort. Assume that $\varphi(\bar{x},b)$ is strongly minimal where $b$ is in $\mathfrak{C}^{\rm eq}$. Then we can express that any Morley sequence $\bar{a}_{0},\ldots,\bar{a}_{\mu}$ of $\varphi(\bar{x},b)$ fulfils $\vDash\psi(\bar{a}_{0},\ldots,\bar{a}_{\mu},b)$. 2. b)

$X\sim Y$* for strongly minimal sets can be expressed.*

Proof.

b) follows from a) and

[TABLE]

is the desired formula in a). ∎

Definition 12.10**.**

If $X$ is a strongly minimal subset of $\mathfrak{C}^{n}$ and $X\sim\varphi(\bar{x},b)$ where $b\in\mathfrak{C}^{\rm eq}$, then we say that $X$ is encoded by $\varphi(\bar{x},y)$.

We define codes similarly as in [10] and [6]. It is a modification of E. Hrushovski’s definition [15] to our context. The set $\mathcal{C}$ of good codes is a modification of $\mathcal{X}$.

Definition 12.11**.**

$\varphi_{\alpha}(\bar{x},{y})$ is a code formula or short a code, if it fulfils the following conditions:

1. a)

Length$(\bar{x})=n_{\alpha}\geq 2$, $y$ is an element in $\mathfrak{C}^{eq}$ that encodes a substructure $\langle\bar{y}\rangle$, and $\bar{x}=\bar{x_{1}}\bar{x_{2}}$ is an o-system over $\langle\bar{y}\rangle$, where $\bar{x_{1}}$ is in $U_{1}$ and $\bar{x_{2}}$ is in $U_{2}$. 2. b)

The set $\varphi_{\alpha}(\bar{x},b)$ is either empty or strongly minimal. 3. c)

$n_{\alpha}$ is the o-dimension for all solutions. 4. d)

$\varphi_{\alpha}(\bar{x},b)\sim\varphi_{\alpha}(\bar{x},b^{\prime})$ implies $b=b^{\prime}$. 5. e)

If some non-empty $\varphi_{\alpha}(\bar{x},b)$ is groupless, then all $\varphi_{\alpha}(\bar{x},b^{\prime})$ are. 6. f)

$\varphi_{\alpha}(\bar{x_{1}}+\bar{m_{1}},\bar{x_{2}}+\bar{m_{2}},b)$ is encoded by $\varphi_{\alpha}$ for all $\bar{m_{1}}\in\mathfrak{C}_{1}^{\mid\bar{x_{1}}\mid}$ and all $\bar{m_{2}}\in\mathfrak{C}_{2}^{\mid\bar{x_{2}}\mid}$. 7. g)

For all o-transformations $H$ over $\langle\bar{d}\rangle$ the set $\varphi_{\alpha}(H(\bar{x}),,b)$ is encoded by $\varphi_{\alpha}$.

By b) and d) $b$ is the canonical parameter of $\varphi_{\alpha}(\bar{x},b)$.

Lemma 12.12**.**

There is a set ${\mathcal{C}^{0}}$ of codes $\varphi_{\alpha}(\bar{x},y)$ that encodes the strongly minimal sets that are encoded by the formulas in ${\mathcal{X}}$.

Proof.

The proof is similar as the proof in [6] and that proof comes from [10]. The formulas in ${\mathcal{X}}$ have the properties a)–d). Using Lemma 12.6 we can assume w.l.o.g. that the formulas in ${\mathcal{X}}$ satisfy a)–e). Since ${\mathcal{X}}$ is closed under suitable affine o-transformations given in f) and g) by compactness there are finitely many $\varphi_{1},\ldots,\varphi_{r}$ in ${\mathcal{X}}$ that encode all possible suitable affine o-transformations of some given $\varphi(\bar{x},b)$. Moreover we know that either all or none encode groupless sets by Lemma 12.5.

Choose a sequence $w_{1},\ldots,w_{r}$ of different definable elements in $T^{\rm eq}$. Define

[TABLE]

Finally let $\varphi_{\alpha}(\bar{x},y_{1},y)=\bigvee\limits_{i=1}^{r}(\varphi^{\prime}_{i}(\bar{x},y)\wedge y_{1}=w_{i})$. $\varphi_{\alpha}$ has the properties a)–e). To show f) and g) let $b$, $\bar{m_{1}},\bar{m_{2}}$, $H_{1},H_{d},H_{2}$ be given. By construction $\varphi_{\alpha}(\bar{x},b_{1},b)$ is equivalent to some $\varphi(H^{\prime}_{1}(\bar{x_{1}})+\bar{m_{1}}^{\prime},H^{\prime}_{2}(\bar{x_{2})}+H^{\prime}_{d^{\prime}}(\bar{x_{1}})+\bar{m_{2}}^{\prime},b^{\prime})$. Hence

[TABLE]

[TABLE]

and the right side is encoded by $\varphi_{\alpha}$ by construction, since

[TABLE]

for some $H_{c}$. ∎

Theorem 12.13**.**

There is a set ${\mathcal{C}}$ of good codes such that for every $\varphi(\bar{x},b)$ in ${\mathcal{X}}$ there is a unique $\varphi_{\alpha}(\bar{x},c)$ in ${\mathcal{C}}$ such that $\varphi(\mathfrak{C},b)\sim\varphi_{\alpha}(\mathfrak{C},c)$.

Proof.

(modification of the proofs in [6] and [10])

Let $\alpha_{i}$ be a list of all codes from $\mathcal{C}^{0}$ (see Lemma 12.12). Again define:

[TABLE]

$\varphi^{\prime}_{\alpha_{i}}$ satisfies a)–e) we have to show f) and g). By construction $\varphi^{\prime}_{\alpha_{i}}(H(\bar{x})+\bar{m},b)$ is encoded by $\varphi_{\alpha_{i}}$. We need only to show that no $\varphi_{\alpha_{j}}$ with $j<i$ encodes it. Suppose that

[TABLE]

Then

[TABLE]

for some $b^{\prime\prime}$. This contradicts the definition of $\varphi^{\prime}_{\alpha}$. Hence ${\mathcal{C}^{+}}=\{\alpha^{\prime}_{i}:i<\omega\}$ has the desired properties. ∎

Corollary 12.14**.**

In the definition of C(1) - C(7) we can replace ${\mathcal{X}}$ by a set ${\mathcal{C}}$ of good codes.

Corollary 12.15**.**

If $D\leq\mathfrak{C}$ and $D^{\prime}$ is a prealgebraic minimal extension of $D$, then there is a unique good code $\alpha$ such that there is some $b$ in ${\rm dcl}^{\rm eq}(D)$ and a generic solution $\bar{a}$ of $\varphi_{\alpha}(\bar{x},b)$ that generates $D^{\prime}$ over $D$. If $\bar{c}$ is a generating o-system of $D^{\prime}$ over $D$, then there is some $d\in{\rm dcl}^{\rm eq}(D)$ such that $\bar{c}$ is a generic solution of $\varphi_{\alpha}(\bar{x},d)$

Proof.

This follows from the properties of good codes, Lemma 12.13, and Corollary  12.14. ∎

For each $\alpha\in{\mathcal{C}}$ we choose a natural number $m_{\alpha}$ such that the existence of $m_{\alpha}$ common solutions of $\varphi_{\alpha}(\bar{x},\bar{b})$ and $\varphi_{\alpha}(\bar{x},\bar{b}^{\prime})$ implies $\varphi_{\alpha}(\bar{x},\bar{b})\sim\varphi_{\alpha}(\bar{x},\bar{b}^{\prime})$. This is possible by the strong minimality of $\varphi_{\alpha}(\bar{x},\bar{y})$.

Theorem 12.16**.**

For each $\alpha\in{\mathcal{C}}$ and $\lambda\geq m_{\alpha}$ there is a formula $\psi_{\alpha}(\bar{x}_{0},\ldots,\bar{x}_{\lambda})$ with the following properties:

1. a)

For any initial segment $\{\bar{e}_{0},\ldots,\bar{e}_{\lambda},\bar{f}\}$ of a Morley sequence of $\varphi_{\alpha}(\bar{x},b)$

[TABLE]

holds. 2. b)

For each realization $(\bar{e}_{0},\ldots,\bar{e}_{\lambda})$ of $\psi_{\alpha}$ there is a unique $b$ with $\vDash\varphi_{\alpha}(\bar{e}_{i},b)$ for $0\leq i\leq\lambda$. Moreover $b\in{\rm dcl}^{\rm eq}(\bar{e}_{i_{1}},\ldots,\bar{e}_{i_{m_{\alpha}}})$ for any $i_{1}<\ldots<i_{m_{\alpha}}$.

$($We call $b$ the canonical parameter of the sequence $\bar{e}_{0},\ldots,\bar{e}_{\lambda})$. 3. c)

$b$* encodes a unique canonical substructure $\langle\bar{b}\rangle$ for a corresponding formula in $\mathcal{X}^{home}$ and each realization of $\psi_{\alpha}$ forms an o-system over $\langle\bar{b}\rangle$.* 4. d)

If $\vDash\psi_{\alpha}(\bar{e}_{0},\ldots,\bar{e}_{\lambda})$, then for $i\in\{0,\ldots,\lambda\}:$

[TABLE] 5. e)

Given a realization $(\bar{e}_{0},\ldots,\bar{e}_{\lambda})$ of $\psi_{\alpha}$ with canonical parameter $b$ as in b), we have the following:

Suppose $\alpha$ is groupless:

If $\bar{e}_{i}$ is a generic solution of $\varphi(\bar{x},\bar{b})$, then \bar{e}_{i}-H\bar{e}_{j}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 566\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 566\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 566\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 566\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}\limits_{\bar{b}}\bar{e}_{i} for all o-transformations $H$ over $\bar{b}$ and $j\neq i$.

Suppose $\alpha$ is a coset code, then:

$\varphi_{\alpha}(\bar{x},b)$* is a group-set.* 2. 3)

$\psi_{\alpha}(\bar{e}_{0},\ldots,\bar{e}_{i-1},\bar{e}_{i}-\bar{e}_{j},\bar{e}_{i+1},\ldots,\bar{e}_{\lambda})$* for $j\neq i$.* 3. 4)

$\psi_{\alpha}(\bar{e}_{0},\ldots,\bar{e}_{i-1},H\bar{e}_{i},\bar{e}_{i+1},\ldots,\bar{e}_{\lambda})$* for all suitable o-transformations $H\in{\rm Inv}(\varphi_{\alpha}(\bar{x},b))$ over $\bar{b}$.* 4. 5)

Moreover, if $\bar{e}_{i}$ is generic in $\varphi_{\alpha}(\bar{x},\bar{b})$, then \bar{e}_{i}-H\bar{e}_{j}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 566\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 566\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 566\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 566\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}\limits_{\bar{b}}\bar{e}_{i} for all $j\neq i$ and all suitable o-transformations $H$ over $\bar{b}$ not in ${\rm Inv}(\varphi_{\alpha}(\bar{x},b))$.

Proof.

(Copy of the corresponding proof in [10] but in another theory.)

We consider the following partial type

[TABLE]

Claim. $\Sigma$ has the properties a) – e).

Proof of the claim. a) is clear.

Given such a realization $\bar{e}_{0},\ldots,\bar{e}_{\lambda}$ of $\Sigma$, where $b^{\prime}$ and $\bar{a}_{0},\ldots,\bar{a}_{\lambda},\bar{f}$ are given as above. Hence $\{\bar{e}_{i}\}_{0\leq i\leq\lambda}$ is a Morley sequence of $\varphi_{\alpha}(\bar{x}+\bar{f},\bar{b}^{\prime})$. Then $\varphi_{\alpha}(\bar{x}+\bar{f},b^{\prime})\sim\varphi_{\alpha}(\bar{x},b)$ for some $b$ by f) in the definition of codes. Since $b$ is the canonical parameter of the generic type determined by $\varphi_{\alpha}(\bar{x},b)$, the sequence $\{\bar{e}_{i}\}_{0\leq i\leq\lambda}$ is a Morley sequence for $\varphi_{\alpha}(\bar{x},b)$. Given another $b^{*}$ which satisfies $\varphi_{\alpha}(\bar{e}_{i},\bar{y})$ for $m_{\alpha}$ many $i$’s, it follows that $\varphi_{\alpha}(\bar{x},b^{*})\sim\varphi_{\alpha}(\bar{x},b)$ by the choice of $m_{\alpha}$. By d) in the code-definition $b^{*}=b$. Hence b) is true for $\Sigma$.

c) is clear.

Since $\bar{a}_{0},\ldots,\bar{a}_{i-1},\bar{f},\bar{a}_{i+1},\ldots,\bar{a}_{\lambda},\bar{a}_{i}$ is again a Morley sequence for $\varphi_{\alpha}(\bar{x},b^{\prime})$ we have

[TABLE]

and hence

[TABLE]

We get d).

To prove e) we assume first that $\alpha$ is groupless (e1). That means $\varphi_{\alpha}(\bar{x},b)$ is not a torsor set. By Lemma 12.8 the assertion follows.

Otherwise $X=\varphi_{\alpha}(\mathfrak{C},b^{\prime})$ is a torsor set. Hence $X-\bar{f}\sim\varphi_{\alpha}(\bar{x},b)$ is a group set since $\bar{f}$ is in $X$. This is (e2).

To prove (e3) we extend the Morley sequence $\{\bar{e}_{i}:0\leq i\leq\lambda\}$ by an element $\bar{d}$. Then

[TABLE]

is again a Morley sequence for $\varphi_{\alpha}(\bar{x},\bar{b})$. Hence

[TABLE]

Similarly we get e4).

e5) follows again by Lemma 12.8. $\Box$(Claim)

Using compactness we get a finite part $\psi^{\prime}_{\alpha}$ of $\Sigma$ that implies a), b), c), e1), e2), e5).

If $\alpha$ is groupless consider the following operations:

[TABLE]

and $\mathcal{V}$ be the subgroup generated by these operations. $\mathcal{V}$ is finite. Then

[TABLE]

satisfies d) and is also part of $\Sigma$.

If $\alpha$ is an coset code, property d) follows from e3) and e4). Hence it is sufficient that $\psi_{\alpha}$ satisfies e3) and e4). Let ${\mathcal{W}}(\bar{x}_{0},\ldots,\bar{x}_{\lambda})$ be the subgroup generated by the operations mentioned in e3) and e4). Again ${\mathcal{W}}$ is finite, and depends on ${\rm Inv}(\varphi_{\alpha}(\bar{x},\bar{b}))$. Note that $\lambda\geq m_{\alpha}$, hence $b$ remains constant in b) after applying these operations. Set therefore:

[TABLE]

which has the required properties. ∎

Definition 12.17**.**

Let $\alpha$, $\lambda$ and $\psi_{\alpha}$ be as above. A realization of $\psi_{\alpha}$ is called a difference sequence for $\alpha$. Moreover, given a realization $\bar{e}_{0},\ldots,\bar{e}_{\lambda}$ of $\psi_{\alpha}$, we denote by a derived difference sequence one obtained by composition of the following operations:

[TABLE]

If $\nu\leq\lambda$ and we use the operations above only for $i\leq\nu$, then we speak about a $\nu$-derived sequence.

Corollary 12.18**.**

A permutation of a difference sequence is a difference sequence.

Proof.

Note that all permutations of a difference sequence are obtained by the operation in d) of Theorem 12.16. ∎

13. Bounds for difference sequences

A difference sequence for a good code $\alpha\in\mathcal{C}$ forms an o-system. See Theorem 12.16c). The next lemma is Lemma 5.1 in [6]. In our contex it needs a modified proof.

Lemma 13.1**.**

For every code formula $\varphi_{\alpha}(\bar{x},y)$ and every natural number $r$, there is some $\lambda(r,\alpha)=\lambda>0$ such that for every $D\leq\mathfrak{C}$ and every difference sequence $\bar{e_{0}},\ldots,\bar{e_{\mu}}$ for $\varphi_{\alpha}(\bar{x},y)$ with canonical parameter $b$ and $\mu\geq\lambda$ either

1. (i)

the canonical parameter of some $\lambda$-derived sequence of $\bar{e_{0}},\ldots,\bar{e_{\mu}}$ lies in $dcl^{eq}(D)$

or

1. (ii)

for every $\alpha$-o-system $\bar{m}$ the sequence $\bar{e_{0}},\ldots,\bar{e_{\mu}}$ contains a subsequence $\bar{e_{i_{0}}},\ldots,\bar{e_{i_{r-1}}}$, such that $m_{\alpha}\leq i_{j}$ and $e_{i_{j}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle DB\bar{e_{i_{0}}},\ldots,\bar{e_{i_{j-1}}}\rangle$, where $B=\langle\bar{m},\bar{e_{0}},\ldots,\bar{e_{m_{\alpha}-1}}\rangle$.

Proof.

For $T_{3}$ the proof in [6] needs some new ideas. We assume that the fixed finite field is $\mathbb{F}(q)$ with $q$ elements. We have $\bar{e_{j}}=\bar{e_{j}^{1}}\bar{e_{j}^{2}}$, where $U_{1}(\bar{e_{j}^{1}})$, $U_{2}(\bar{e_{j}^{2}})$ and analogusly $\bar{x}=\bar{x^{1}}\bar{x^{2}}$. Let $\bar{x^{1}}$ be of length $n_{\alpha,1}$ and $\bar{x^{2}}$ be of length $n_{\alpha,2}$.

If assertion (i) is not true, then every suitable coset of $\mathfrak{C}^{n_{\alpha}}/D^{n_{\alpha}}$ contains at most $m_{\alpha}$-many elements $\bar{e_{i}}$ with $i\leq\lambda$ of the difference sequence under consideration. Otherwise we could substract an $\bar{e_{j}}$ with $j\leq\lambda$ of a larger coset and would get a $\lambda$-derived sequence with the canonical parameter in $D^{eq}$.

Claim Assume $n_{\alpha,1}\not=0$. Then there is a natural number $k_{\alpha}$, such that for every $\mu$ and every $i\leq\mu$ we have $\mid\{j:i\not=j,\bar{e_{j}}^{1}=\bar{e_{i}}^{1}modD\}\mid\leq k_{\alpha}$.

Proof.

If $n_{\alpha,2}=0$, then $k_{\alpha}=m_{\alpha}$ fulfills the assertion.

Now we assume, that $n_{\alpha,2}\not=0$. We choose $i$ such that $\mid\{j:i\not=j,\bar{e_{j}}^{1}=\bar{e_{i}}^{1}modD\}\mid$ is maximal. Since the permutation of a difference sequence is again a difference sequence, we can assume that $i=0$. W.l.o.g. there is some $l$, such that $\bar{e_{0}}^{1}=\bar{e_{l}}^{1}modD$, since otherwise we have only to ensure $m_{\alpha}\leq k_{\alpha}$ for this case.

By (d) in the definition of a difference sequence

[TABLE]

is again a difference sequence.

Hence w.l.o.g. $\bar{e_{0}}^{1}\in D$ and we consider $Z=\{j:\bar{e_{j}}^{1}\in D\}$. Our aim is, that we can find $k_{\alpha}$ such that, $\mid Z\mid\leq k_{\alpha}$ for all possible $Z$. By assumption there are at most $m_{\alpha}$-many elements $e_{j}^{2}$ in the same coset modulo $D$.

After some permutations we can assume that there is some $\beta$, such that

[TABLE]

We consider $E(\beta)_{2}=\langle\bar{e_{0}}^{2},\ldots,\bar{e_{\beta}^{2}}\rangle$. Let $s_{2}$ be $ldim(E(\beta)_{2}/B_{2}+D_{2})$, where $B=\langle\bar{m},\bar{e_{0}},\ldots,\bar{e_{m_{\alpha}-1}}\rangle$, as above. Then $ldim(E(\beta)_{2}/D)\leq s_{2}+(m_{\alpha}+1)n_{\alpha,2})$. All $\bar{e_{j}}^{2}$ with $j\leq\beta$ are in the finite vector space $E(\beta)_{2}/D_{2}$ and at most $m_{\alpha}$ many are in the same coset of $D_{2}$. We get

[TABLE]

Now define

[TABLE]

Then $s_{2}\leq\mid Y\mid n_{\alpha,2}$ and hence

[TABLE]

Now we choose $k_{\alpha}$, such that for all $\beta>k_{\alpha}$ we have $\delta(B/D)<\mid Y\mid$. By (C 5b) there is at most one $\bar{e_{i}}$ that is weakly generic over $\langle DB\bar{e_{0}},\ldots\bar{e_{i-1}}\rangle$. This is not possible, since $\bar{e_{i}}^{1}\in D$.

∎

Case $n_{\alpha,1}\not=0$.

We define $E=E(\lambda)=\langle\bar{e_{0}},\ldots,\bar{e_{\lambda}}\rangle$. Let $s$ be $ldim(E_{1}/B_{1}+D_{1})$. Then $ldim(E_{1}/D_{1})\leq s+(m_{\alpha}+1)n_{\alpha,1}$.Then

[TABLE]

Define $X=\{i:m_{\alpha}\leq i\leq\lambda,\bar{e_{i}^{1}}\notin\langle D_{1}B_{1}+\langle\bar{e_{0}^{1}},\ldots,\bar{e_{i-1}^{1}}\rangle_{1}\rangle$. Since $s\leq\mid X\mid n_{\alpha,1}$, we get

[TABLE]

If we choose $\lambda$ sufficiently large we get

[TABLE]

By (C 5b) we get $\bar{e_{i_{0}}},\ldots,\bar{e_{i_{r-1}}}$, such that $m_{\alpha}<i_{j}$ and $e_{i_{j}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle DB\bar{e_{i_{0}}},\ldots,\bar{e_{i_{j-1}}}\rangle$

Case $n_{\alpha,1}=0$.

If $E=E(\lambda)=\langle\bar{e_{0}},\ldots,\bar{e_{\lambda}}\rangle$ as above, then $E=\langle E_{2}\rangle$. We define $s=ldim(E_{2}/B_{2}+D_{2})$. Then $ldim(E_{2}/D_{2})=s+(m_{\alpha}+1)n_{\alpha,2}$ and

[TABLE]

Similarly as above we define $X=\{i:m_{\alpha}\leq i\leq\lambda,\bar{e_{i}^{2}}\notin\langle D_{2}+B_{2}+\langle\bar{e_{0}^{2}},\ldots,\bar{e_{i-1}^{2}}\rangle_{1}\rangle\}$. Since $s\leq\mid X\mid n_{\alpha,2}$ we get

[TABLE]

We can choose $\lambda$ as large such that

[TABLE]

By (C 5b) we get $\bar{e_{i_{0}}},\ldots,\bar{e_{i_{r-1}}}$, such that $m_{\alpha}<i_{j}$ and $e_{i_{j}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle DB\bar{e_{i_{0}}},\ldots,\bar{e_{i_{j-1}}}\rangle$

∎

Now we consider all finite-to-one functions $\mu^{*}$ and $\mu$ defined on the good codes $\alpha\in{\mathcal{C}}$ with values in $\mathbb{N}$. We assume that the following inequalities hold:

• •

$\mu(\alpha)\geq m_{\alpha}$,

• •

$\mu^{*}(\alpha)\geq\max(\lambda(m_{\alpha}+1,\alpha)+1,n_{\alpha}+1)$,

• •

$\mu(\alpha)\geq\lambda(\mu^{*}(\alpha),\alpha)+1$.

For the definition above we fix a function $\lambda(r,\alpha)$ given by Lemma 13.1 and we assume that it is monotonous in the first argument.

Finally we will get for each such function $\mu$ as above a ”generic” substructure $P^{\mu}(\mathfrak{C})\leq\mathfrak{C}$ such that for all good codes $\alpha\in\mathcal{C}$ there is no difference sequences of length $\mu(\alpha)+1$.

We will extend the language $L$ by a new predicate $P^{\mu}$ and consider the structure $\langle\mathfrak{C},P^{\mu}(\mathfrak{C})\rangle$ in the new language $L^{\mu}$. $P^{\mu}(\mathfrak{C})$ will be the desired $L$-structure of finite Morley rank. We will get $P^{\mu}(\mathfrak{C})$ by amalgamation of strong subspaces in the class $\mathcal{K}^{\mu}_{\rm fin}$ defined below.

Definition 13.2**.**

Let $\mathcal{K}^{\mu}$ be the class of all strong substructures $U$ of $\mathfrak{C}$ in $\mathcal{H}_{3}$, such that for every good code $\alpha$ there is no difference sequence for $\alpha$ of length $\mu(\alpha)+1$ in $U$. $\mathcal{K}_{\rm fin}^{\mu}$ are the finite structures in $\mathcal{K}^{\mu}$.

Note that difference sequences are given by realizations of the formulas$\psi_{\alpha}(\bar{x}_{0},\ldots,\bar{x}_{\mu(\alpha)})$ in Theorem 12.16.

Here $\mathcal{K}^{\mu}$ is a class of substructures in $\mathcal{H}_{3}$. In [6] we work with subspaces of $\mathfrak{C}_{1}$. We hav e to modify the proofs of [6] for the new contexts. Let $D\leq D^{\prime}$ be strong subspaces of $\mathfrak{C}$ in $\mathcal{H}_{3}$ with ${\rm ldim}(D^{\prime}/D)$ finite. By Lemma 9.7 there is a sequence of minimal strong extensions for starting with $D$ and arriving $D^{\prime}$. We call it a geometrical sequence. In the next lemmas we will investigate the minimal steps in this sequence, especially the prealgebraic minimal steps for the case that $D\in\mathcal{K}^{\mu}$ but $D^{\prime}\notin\mathcal{K}^{\mu}$.

Lemma 13.3**.**

*Assume $D\leq D^{\prime}\leq\mathfrak{C}$, $D\in\mathcal{K}^{\mu}$, $D^{\prime}$ is a prealgebraic minimal extension of $D$ and $D^{\prime}$ is not $\mathcal{K}^{\mu}$. Let $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)}$ be a difference sequence in $D^{\prime}$ for a good code $\alpha$, such that its canonical parameter $c$ is in ${\rm dcl}^{\rm eq}(D)$. Then we find a difference sequence $\bar{d}_{0},\ldots,\bar{d}_{\mu(\alpha)}$ for $\alpha$ in $D^{\prime}$ with the same canonical parameter such that $\bar{d}_{0},\ldots,\bar{d}_{\mu(\alpha)-1}$ are in $D$, and $\bar{d}_{\mu(\alpha)}$ is a $D$-generic realization of $\varphi_{\alpha}(\bar{x},\bar{c})$ that generates $D^{\prime}$ over $D$.

If we cannot find the new sequence by a permutation of the old one, then $\alpha$ is a group code and the new sequence is obtained using operations as $\bar{e}_{j}$ is replaced by some $H(\bar{e}_{j})-\bar{e}_{i}$ where $H$ is in ${\rm Inv}(\varphi_{\alpha}(\bar{x},\bar{c}))$. $\alpha$ is the unique good code that describes $D^{\prime}$ over $D$.*

Proof.

Let $\bar{c}$ be a generating o-system of the canonical base algebra coded by $c$. Since $D\in\mathcal{K}^{\mu}$, there is some $\bar{e}_{i}$ not completely in $D$. Since $D\leq\mathfrak{C}$ by C(3) $\bar{e}_{i}$ is $D$-generic and generates $D^{\prime}$ over $D$. If there is some other $\bar{e}_{j}$ not completely in $D$, then again $\bar{e}_{j}$ is $D$-generic and generates $D^{\prime}$ over $D$. By Lemma 11.2 $\bar{e}_{i}=H(\bar{e}_{j})-\bar{m}_{j}$ where $H$ is a suitable o-transformation over $\bar{c}$ and $\bar{m}_{j}$ is in $D$. Then $H(\bar{e}_{j})-\bar{e}_{i}$ is in $D$. Since $\bar{e}_{j}$ is $D$-generic, we have

[TABLE]

By the properties of a difference sequence it follows that $\alpha$ is a group code and $H$ is in ${\rm Inv}(\varphi_{\alpha}(\bar{x},\bar{c}))$. If we replace $\bar{e}_{j}$ by $H(\bar{e}_{j})-\bar{e}_{i}$ we obtain again a difference sequence with the same canonical parameter and this sequence has one more element in $D$. We can iterate the argument to obtain the assertion.

Finally for $D^{\prime}$ over $D$ there exists a unique code in $\mathcal{C}$ by Theorem 12.13. ∎

Corollary 13.4**.**

Let $D$ be in $\mathcal{K}^{\mu}$ and $D\leq D^{\prime}\leq\mathfrak{C}$ be a minimal extension. If $D^{\prime}$ has o-dimension one over $D$, then $D^{\prime}$ is in $\mathcal{K}^{\mu}$. Otherwise, in the prealgebraic case, $D^{\prime}$ is in $\mathcal{K}^{\mu}$ if and only if none of the following two conditions holds:

1. a)

There is a code $\alpha\in{\mathcal{C}}$ and a difference sequence $\bar{e}_{0},\ldots\bar{e}_{\mu(\alpha)}$ for $\alpha$ in $D^{\prime}$ such that

1. i)

$\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)-1}$* are contained in $D$.* 2. ii)

$D^{\prime}=\langle D\bar{e}_{\mu(\alpha)}\rangle^{\ell}$. 3. iii)

In this case $\alpha$ is the unique good code that describes $D^{\prime}$ over $D$. 2. b)

There exists a code $\alpha\in{\mathcal{C}}$ and a difference sequence for $\alpha$ in $D^{\prime}$ of length $\mu(\alpha)+1$ with canonical parameter $b$ and with a subsequence $\bar{e}_{0},\ldots,\bar{e}_{\mu^{*}(\alpha)-1}$ of length $\mu^{*}(\alpha)$ such that $\bar{e}_{i}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $D+B+\langle\bar{e}_{0},\ldots,\bar{e}_{i-1}\rangle$ where $B$ is generated by the first $m_{\alpha}$ elements of the given difference sequence.

Proof.

Consider first the case where ${\rm o-dim}(D^{\prime}/D)=1$. Assume that $D^{\prime}$ is not in $\mathcal{K}^{\mu}$. That means there is a difference sequence $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)}$. If the canonical parameter $b$ lies in ${\rm dcl}^{\rm eq}(D)$, then all $\bar{e}_{i}$ would be in $D$, since $\bar{e}_{i}\notin D$ implies, that it generates a strong minimal extension over $D$ with an o-dimension over $D$ greater than 1. This contradicts $D\in\mathcal{K}^{\mu}$.

Otherwise by Lemma 13.1 some $\bar{e}_{j}$ is a realization of $\varphi_{\alpha}(\bar{x},b)$ and an o-system over $D$ since $\mu(\alpha)\geq\lambda(1,\alpha)$. Again we have a contradiction.

Finally we assume that $D^{\prime}$ is minimal prealgebraic over $D$. Again we assume that there is a difference sequence $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)}$ in $D^{\prime}$ for some good code formula $\varphi_{\alpha}(\bar{x},b)$ where $b$ is the canonical parameter. If $b$ lies in ${\rm dcl}^{\rm eq}(D)$, then by Lemma 13.3 we get case a). Otherwise since $\mu(\alpha)\geq\lambda(\mu^{*}(\alpha),\alpha)$ our sequence contains a subsequence of length $\mu^{*}(\alpha)$ as described in b) by Lemma 13.1. ∎

14. Amalgamation in $\mathcal{K}^{\mu}$

Again we work in $T_{3}$ or $T_{3}^{\rm eq}$.

Lemma 14.1**.**

Let $B\subseteq A$ and $B\subseteq C$ in $\mathcal{H}_{3}$ all be strong subspaces of $\mathfrak{C}$. Assume that $A$ and $C$ are minimal prealgebraic extensions of $B$, $A\cap C=B$, and that $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)}$ is a difference sequence for a good code $\alpha$ in $\langle AC\rangle$. Then there is a derived difference sequence of the above sequence with the canonical parameter in ${\rm dcl}^{\rm eq}(C)$ or in ${\rm dcl}^{\rm eq}(A)$.

Proof.

By assumption $\langle AC\rangle=A\otimes_{B}C\leq\mathfrak{C}$, since $A,B,C$ are strong substructures and $\delta(A)=\delta(B)=\delta(C)$. We assume that the assertion of the lemma is not true. Let $E=\langle\bar{e}_{0},\ldots,\bar{e}_{m_{\alpha}-1}\rangle$. By Lemma 13.1 we get a subsequence $\bar{e}_{i_{0}},\ldots,\bar{e}_{i_{\mu^{*}(\alpha)}}$, such that $\bar{e}_{i_{j}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle CE\bar{e}_{i_{0}},\ldots,\bar{e}_{i_{j-1}}\rangle$. Since $\mu^{*}(\alpha)\geq\lambda(m_{\alpha}+1,\alpha)+1$ we get a subsequence of this sequence of length $m_{\alpha}+1$ such that every element is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle CE\rangle$ and over $\langle AE\rangle$. Again we have applied Lemma 13.1. By C(6) $\varphi_{\alpha}(x,y)$ defines a torsor set and by the properties of a difference sequence a group set. Hence by C(6) $\bar{b}\in{\rm dcl}^{\rm eq}(B)$, a contradiction to the assumption. ∎

By definition all structures $A$ in $\mathcal{K}^{\mu}$ are in $\mathcal{H}_{3}$ and there are strong embeddings of them into $\mathfrak{C}$. For a strong substructure $A$ of $\mathfrak{C}$   ${\rm tp}^{\mathfrak{C}}(A)$ is given by the isomorphism type of $A$. Embeddings of strong substructures with strong images are elementary embeddings.

Theorem 14.2**.**

The class $\mathcal{K}_{\rm fin}^{\mu}$ has the amalgamation property with respect to strong embeddings: If $B\leq C\leq\mathfrak{C}$, $B^{\prime}\leq A\leq\mathfrak{C}$, $B\cong B^{\prime}$, and $B,B^{\prime},A,C$ are in $\mathcal{K}^{\mu}_{fin}$, then there is some $C\leq D\leq\mathfrak{C}$ in $\mathcal{K}^{\mu}_{fin}$, such that the strong isomorphism of $B^{\prime}$ onto $B$ can be extended to an strong embedding of $A$ in $D$.

Proof.

Since $\mathfrak{C}$ is rich we can assume w.l.o.g., that $B=B^{\prime}$ and can exchange the roles of $C$ and $A$. Splitting $A$ and $C$ into chains of minimal strong extensions in $\mathcal{K}^{\mu}$ (C4) we can assume w.l.o.g. that $A$ and $C$ are minimal strong extensions.

Case 1:   ${\rm o-dim}(C/B)=1$ or ${\rm o-dim}(A/B)=1$.

W.l.o.g. $A=\langle Ba\rangle$. If $a$ is algebraic and a solution of a divisor problem of $B$, that has also a solution in $C$, then $D=C$. Otherwise $D=\langle Ca\rangle$ is an amalgam in $\mathcal{K}$. By Corollary 13.4 the amalgam $D$ is in $\mathcal{K}^{\mu}$.

Case 2: Both extensions $C/B$ and $A/B$ are prealgebraic.

Then $D=C\otimes_{B}A$ exists and is in $\mathcal{K}$ by Theorem 6.15. We assume that $D$ is not in $\mathcal{K}^{\mu}$ and show in this case that $C$ and $A$ have the same type over $B$. There is a good code $\alpha$ with a difference sequence $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)}$ in $D$. By Lemma 14.1 and symmetry we may assume w.l.o.g. that its canonical parameter $\bar{b}$ lies in ${\rm dcl}^{\rm eq}(C)$. By Lemma 13.3 we may assume that $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)-1}$ are in $C$ and $\bar{e}_{\mu(\alpha)}$ is an $C$-generic realization of $\varphi_{\alpha}(\bar{x},\bar{b})$ which generates $D$ over $C$.

Case 2A: We first assume that the canonical parameter for some $(\mu(\alpha)-1)$-derived difference sequence is in ${\rm dcl}^{\rm eq}(B)$. Since this difference sequence has the same properties, we denote it again by $\bar{e}_{0},\ldots,\bar{e}_{\mu(\alpha)}$. By Lemma 10.1 we have the following two subcases:

Case 2.A.1:   $\bar{e}_{\mu(\alpha)}\in A$.

By minimality of $A$ over $B$ we have $A=\langle B\bar{e}_{\mu(\alpha)}\rangle$. Since $A$ is in $\mathcal{K}^{\mu}$, there exists an $\bar{e}_{i}$ that lies in $C$ and not in $B$. By Lemma 10.1 $B\leq C$ implies that $\bar{e_{i}}$ is $B$-generic and therefore isomorphic to $\bar{e}_{\mu(\alpha)}$ over B. Furthermore $C=\langle\bar{e}_{i}B\rangle$.

Case 2.A.2:   $\bar{e}_{\mu(\alpha)}$ is a solution of $\varphi_{\alpha}(\bar{x},b)$ generic over $A$.

We have $D=C\otimes_{B}A$. The canonical base algebra of $\varphi_{\alpha}(\bar{x},b)$ is in $B$. By Lemma 10.4 we get $D=C\otimes_{B}\langle B\bar{e}_{\alpha}\rangle$. Hence we get $\bar{e}_{\mu(\alpha)}=\bar{a}+\bar{m}$, where $\bar{m}$ is in $C$ and $\bar{a}$ is an o-system that generates $A$ over $B$. It is the solution of a code formula with parameter in $B$. Then $\bar{e}_{\mu(\alpha)},\bar{a},\bar{m}$ is a $B$-independent tripel. By Lemma 12.4 $\bar{e}_{\mu(\alpha)},\bar{a},\bar{m}$ are generic elements of cosets of an $acl(B)$-definable group. But then the group is defined by $\varphi(\bar{x},b)$ by the properties of a derived sequence. $\bar{a}$ and $-\bar{m}$ are in the same coset. Therefore the have the same type over $B$.

Case2.B:  No $(\mu(\alpha)-1)$-derived sequence has a canonical basis in $dcl^{eq}(B)$.

Let $E$ be $\langle B\bar{e}_{0},\ldots,\bar{e}_{m_{\alpha}-1}\rangle$. Then $b\in dcl^{eq}(E)$. Furthermore

[TABLE]

and

[TABLE]

where $\bar{a}$ is a generating o-system of $A$ over $B$. It is the solution of a code formula with canonical parameter $c$ in $B^{eq}$. By Lemma 11.2 $\bar{a}=H(\bar{e}_{\mu(\alpha)})+\bar{m}$, where $\bar{m}$ is in $C$ and $H(\bar{e}_{\mu(\alpha)})$ is an o-transformation over $E$.

Since $\mu(\alpha)$ is suficciently large, Lemma 13.1 provides us some $\bar{e}_{i}$ weakly generic over $\langle E\bar{m}\rangle$. Let $f$ be an isomorphism, that fixes $\langle E\bar{m}\rangle$ and sends $\bar{e}_{\mu(\alpha)}$ onto $\bar{e}_{i}$. Then $C=\langle Bf(\bar{a})\rangle$ is isomorphic to $A$. ∎

Definition 14.3**.**

Let $D$ be a strong substructure of $\mathfrak{C}$ in $\mathcal{H}_{3}$, that is in $\mathcal{K}^{\mu}$. $D$ is called $\mathcal{K}^{\mu}$-rich if for every finite $B\leq A$ in $K^{\mu}$ with $B\leq D$, there exists an $A^{\prime}$ with $B\leq A^{\prime}\leq D$ and $A$ and $A^{\prime}$ are isomorphic over $B$. Then $A^{\prime}\leq\mathfrak{C}$.

We can use Theorem 14.2 to produce a countable $\mathcal{K}^{\mu}$-rich strong substructure $P^{\mu}(\mathfrak{C})$ in $\mathfrak{C}$ via a Fraïssé-style-argument (see [22]):

Corollary 14.4**.**

There is a countable $\mathcal{K}^{\mu}$-rich strong substructure $P^{\mu}(\mathfrak{C})$ in $\mathfrak{C}$. It’s isomorphism type is unique. Hence it’s type in $\mathfrak{C}$ is uniquely determined.

Corollary 14.5**.**

Let $D$ be a $\mathcal{K}^{\mu}$-rich strong substructure of $\mathfrak{C}$. Then

1. a)

${\rm acl}(D)=D$. 2. b)

$d(D)\geq\aleph_{0}$.

Proof.

a) Let $A$ be a finite subspace of $D$ and $a\in{\rm acl}(A)$. W.l.o.g. we assume that $A$ is a strong subspace of $\mathfrak{C}$ since $D$ is a strong subspace. By property C(2) and Corollary 13.4 every extension $A^{\prime}$ of $A$ by an element algebraic over $A$ is strong and in $\mathcal{K}^{\mu}$. By $\mathcal{K}^{\mu}$-richness follows the assertion.

b) Let $U\leq\mathfrak{C}$ be a maximal substructure of $D$ generated by geometrically independent elements. Then $U$ is strong in $\mathfrak{C}$ and $U$ is in $\mathcal{K}^{\mu}$ (Axiom C(2) ). $U$ cannot be finite since in this case an extension $U^{\prime}$ of $U$ by an geometrically independent element would be in $\mathcal{K}^{\mu}$ and had to be realized in $D$ (Corollary 13.4). ∎

15. The theory $T^{\mu}_{3}$

Definition 15.1**.**

We extend our language $L$ by a predicate $P^{\mu}$. Let $L^{\mu}$ be the extended language. We are interested in $L^{\mu}$-structures $(M,D)$ where $M\preceq\mathfrak{C}$ is a rich model of $T_{3}$, $D$ is a $\mathcal{K}^{\mu}$-rich substructure of $M$, and $d(M/D)\geq\aleph_{0}$. Then $D\leq M$ by definition. The interpretation of $P^{\mu}$ is $D$. We call such a $L^{\mu}$-structure a $2\times$rich $L^{\mu}$-structure. Let $R^{\mu}(M)$ be $R(M)\cap P^{\mu}(M)$. We will show that the theory $T^{\mu}_{3}$ of all $2\times$rich $L^{\mu}$-structures is complete. Later we will also use $M,N,\ldots$ for $L^{\mu}$-structures.

The $L$-reduct of a $2\times$rich $L^{\mu}$-structure is rich in the sense of $T_{3}$. We use ${\rm acl}^{L}$ and often acl only for the algebraic closure in the $L$-reducts. ${\rm acl}^{\mu}$ is the algebraic closure in the full $L^{\mu}$-structure.

Lemma 15.2**.**

We consider a $2\times$rich $L^{\mu}$-structure $(M,D)$ as above. Then

[TABLE]

.

Hence $P^{\mu}$ is definable using $R^{\mu}$. Corollary 14.4 provides us a $2\times$rich $L^{\mu}$-structure. For a code formula $\varphi(\bar{x},y)\in\mathcal{C}$ we use often the coressponding $L$-formula and denote it by $\varphi(\bar{x},\bar{y})\in\mathcal{C}$

Lemma 15.3**.**

Let $(M,P^{\mu}(M)$ be a $L^{\mu}$-structure, where $M$ is a model of $T_{3}$, $P^{\mu}(M)\leq M$ and $P^{\mu}(M)\in\mathcal{K}^{\mu}$. Furthermore let $\varphi_{\alpha}(\bar{x},\bar{b})$ be a code formula. Then $\varphi_{\alpha}(\bar{x},\bar{b})$ has only finitely many solutions in $P^{\mu}(M)$.

Proof.

We assume that there are infinitely many solutions. Choose a finite strong subspace $B\leq M$ in $\mathcal{H}_{3}$ such that $\bar{b}\in{\rm dcl}^{\rm eq}(B)$. There is a solution $\bar{e_{0}}$ in $P^{\mu}(M)$, but not in $B$. By C(3) it is a generic solution over $B$. Since $\delta(\langle B\bar{e_{0}}\rangle)=\delta(B)$ we have $\langle B\bar{e_{0}}\rangle\leq M$. Then we choose a solution $\bar{e_{1}}\in P^{\mu}\setminus\langle B\bar{e_{0}}\rangle$. Again it is a solution generic over $\langle B\bar{e_{0}}\rangle$ by C(3). Again $\delta(\langle B\bar{e_{0}}\bar{e_{1}}\rangle)=\delta(B)$ and therefore $\langle B\bar{e_{0}}\bar{e_{1}}\rangle\leq M$. In this way we get an infinite $L$-Morley-sequence for $\varphi_{\alpha}(\bar{x},\bar{b})$ over $B$ inside $P^{\mu}$. By substraction by one solution we get an infinite diffenrence sequence inside $P^{\mu}$, a contradiction. ∎

Definition 15.4**.**

Let $(M,P^{\mu}(M))$ be a $2\times$rich $L^{\mu}$-structure. and $A$ be a finite substructure of $M$. $A$ satisfies the condition $(*)$ if

$(*)$      $A\leq M$, $A\in\mathcal{H}_{3}$, $A\cap P^{\mu}(M)\in\mathcal{H}_{3}$, and $d(A/A\cap P^{\mu}(M))=d(A/P^{\mu}(M))$.

The condition $d(A/A\cap P^{\mu}(M))=d(A/P^{\mu}(M))$ says that the union of every geometrical basis of $A\cap P^{\mu}(M)$ and every geometrical basis of $A$ over $P^{\mu}(M)$ is a geometrical basis of $A$. This implies

[TABLE]

Note that Lemma 12.1 implies for $A$ with property $(*)$ that

[TABLE]

Then $A\cap cl(P^{\mu}(M))\in\mathcal{H}_{3}$, since it is obtained by a sequence of algebraic and prealgebraic minimal strong extensions over $A\cap P^{\mu}(M)$.

Lemma 15.5**.**

Let $(M,P^{\mu}(M))$ be a $2\times$rich $L^{\mu}$-structure. Assume $A\subseteq M$ satiesfies $(*)$ and $\bar{a}\subseteq M$. Then there is some $D$ with $A\subseteq D$, $\bar{a}\subseteq D$, and $D$ satiesfies $(*)$. Furthermore there is a sequence of minimal strong $\mathcal{H}_{3}$ - extensions

[TABLE]

such that   they satisfy (), $A_{j}=\langle AB_{j}\rangle$ for $j\leq i_{0}$, where $B_{0}=A\cap P^{\mu}(M)$ , $B_{i_{0}}=D\cap P^{\mu}(M)$, and $B_{0}\leq B_{1}\leq\ldots\leq B_{i_{0}}$ is a sequence of minimal strong extension.

Furthermore $A_{j}=\langle AC_{j}\rangle$ for $i_{0}<j\leq i_{1}$, where

$D\cap P^{\mu}(M)\leq C_{i_{0}+1}\leq\ldots\leq C_{i_{1}}=D\cap cl(P^{\mu}(M)$ is a sequence of minimal strong extensions.*

Proof.

Choose $A\leq A^{\prime}\leq M$, such that $\bar{a}\subseteq A^{\prime}$ and $A^{\prime}\in\mathcal{H}_{3}$. Extend $A^{\prime}$ to $D\subseteq cl(A^{\prime})$, such that $D\cap P^{\mu}(M)\subseteq\langle D\cap R^{\mu}(M)\rangle$ and $D\cap cl(R^{\mu}(M))\subseteq cl(D\cap R^{\mu}(M))$. Then $D\in\mathcal{H}_{3}$, $D\cap P^{\mu}(M)\in\mathcal{H}_{3}$, and $D\cap cl(R^{\mu}(M))\in\mathcal{H}_{3}$.

Furthermore $d(D/P^{\mu}(M)=d(D/P^{\mu}(M)\cap D)$. Hence $D$ satiesfies $(*)$. By Lemma 12.1

[TABLE]

We get the desired sequence

[TABLE]

of minimal strong extensions, as described in the Lemma, by (C4). By Lemma 12.1 we have

[TABLE]

By C(4) we get the complete sequence. ∎

Theorem 15.6**.**

Let $M$ and $N$ be $2\times$rich $L^{\mu}$-structures. Assume $A\leq M$ and $f(A)\leq N$ satisfy $(*)$ where $f$ is an $L^{\mu}$-isomorphism of $A$ onto $f(A)$.Then $(M,A)$ and $(N,f(A))$ are $L_{\infty,\omega}^{\mu}$-equivalent.

Corollary 15.7**.**

The $L^{\mu}$-theory $T^{\mu}_{3}$ of the $2\times$rich $L^{\mu}$-structures is complete.

Corollary 15.8**.**

Let $M$ and $N$ be $2\times$rich $L^{\mu}$-structures, $\bar{a}\in R^{\mu}(M)$ and $\bar{b}\in R^{\mu}(N)$. If ${\rm tp}_{L}^{M}(\bar{a})={\rm tp}_{L}^{N}(\bar{b})$, then $(M,\bar{a})$ and $(N,\bar{b})$ are $L_{\infty,\omega}^{\mu}$-equivalent.

Proof.

Adding elements from the algebraic closures we can assume w.l.o.g. that $\langle\bar{a}\rangle$ and $\langle\bar{b}\rangle$ are strong subspaces in $\mathcal{H}_{3}$. Then they fulfil $(*)$. ∎

Proof of Theorem 15.6.

Note that the assumptions imply by Corollary 7.7, that $f$ is an elementary map with respect to the L-reducts of $M$ and $N$. $f$ preserves the geometrically situation.

We show that the conditions in the Theorem describe a winning strategy for the Ehrenfeucht-Fraïssé-game between $(M,A)$ and $(N,f(A))$: Since $M=\langle R(M)\rangle$ and $N=\langle R(N)\rangle$ we can assume w.l.o.g. that the players choose only elements in $R(M)$ and $R(N)$. The situation is completely symmetric. Hence we can assume that player I has choosen some element $a$ in $R(M)$. We show that there are $A\cup\{a\}\subseteq D\leq M$ and $g$ extending $f$ such that again:

$(**)$         

$D\leq M$ and $g(D)\leq N$ fulfil $(*)$, where g is a $L^{\mu}$-isomorphism.

$(**)$ describes again the winning strategy of player II in our Fraïssé-Ehrenfeucht-game for $(M,A)$ and $(N,f(A))$.

We apply Lemma 15.5 and get the desired D with a sequence of minimal strong extensions between $A$ and $D$. We use the notation of that Lemma. Let $g_{0}$ be $f$. We show by induction on i:

If there is a $L^{\mu}$-isomorphism $g_{i}$ of $A_{i}$ into $N$, that extends $f$ and $(**)$ is true,

then we can extend $g_{i}$ to $g_{i+1}$ that sends $A_{i+1}$ into $N$ and fulfils $(**)$. Note that $g_{i}$ is an elementary partiell $L$-isomorphism. $g_{i}$ is not only a $L^{\mu}$-isomorphism. It repects also $cl(P^{\mu})$: For $i_{0}<i$ $A_{i}$ contains for every $b\in cl(P^{\mu}(M)\cap A_{i})=cl(A_{i_{0}})$ a sequence of minimal strong extensions over $A_{i_{0}}$ that contains $b$.

1. (1)

$i<i_{0}$.

1. (a)

$A_{i+1}=\langle A_{i}b\rangle$, $b\in R^{\mu}(M)$, and $d(b/A)=1$.

By Corollary 14.5 there is some $c\in R^{\mu}(N)$ such that $d(c/g_{i}(A_{i})=1$ and with $g_{i+1}(b)=c$ we can extend $g_{i}$ to an $L^{\mu}$-isomorphism $g_{i+1}$. Then $g(A_{i+1})$ satisfies $(*)$. 2. (b)

$A_{i+1}=\langle A_{i}b\rangle$, $b\in R^{\mu}(M)$, $d,e\in B_{i}$, and $[b,d]=e$.

Since $acl(P^{\mu}(N))=P^{\mu}(N)$, we get $g_{i+1}(b)\in P^{\mu}(N)$. 3. (c)

$A_{i+1}=\langle AB_{i}\bar{b}\rangle$, where $\bar{b}\in R^{\mu}(M)$ gives a prealgebraic minimal strong extension of $B_{i}$ by a solution of a formula $\psi(\bar{x},u)$ where $\psi$ corresponds to a good code and $u$ is a canonical base algebra in $B_{i}$.

Since $N$ is $2\times$rich there is a solution $g_{i+1}(b)$ for $\psi(\bar{x},g_{i}(u))$ over $g_{i}(A_{i})$ in $P^{\mu}(N)$. We get $(**)$. 2. (2)

$i_{0}\leq i<i_{1}$.

In this case we have no trancendental minimal strong extensions by $(**)$.

1. (a)

$A_{i+1}=\langle A_{i}b\rangle$, $b\in cl(P^{\mu}(M))\setminus P^{\mu}(M)$, $[b,d]=e$, and $d,e\in cl(P^{\mu}(M))\cap A_{i}$.

There is some $c\in cl(P^{\mu}(N))$ with $[c,g_{i}(d)]=g_{i}(e)$, since $cl(P^{\mu}(N))$ is algebraically closed. If $c\in P^{\mu}(N)$, then

[TABLE]

Furthermore $\delta(g_{i}(B_{i_{0}})=\delta(g_{i}(C_{i}))=\delta(\langle g_{i}(C_{i})c\rangle)$. Since $g_{i}(B_{i_{0}})\leq N$, we get $c\in cl(B_{i_{0}})$. By (+) it follows $c\in acl(g_{i}(B_{i_{0}})$. Then $b\in acl(B_{i_{0}})\subseteq P^{\mu}(M)$, a contradiction. Hence $c\notin P^{\mu}(N)$ and with $g_{i+1}(b)=c$ we get $(**)$ for i+1. 2. (b)

$A_{i+1}=\langle AC_{i}\bar{b}\rangle$, where $\bar{b}\in cl(R^{\mu}(M))$ gives a prealgebraic minimal strong extension of $C_{i}$ by a solution of a formula $\psi(\bar{x},u)$ where $\psi$ corresponds to a good code and $u$ is a canonical base algebra in $C_{i}$, and the solution $\bar{b}$ is an o-system over $P^{\mu}(M)$.

Since $N$ is a model of $T_{3}$, there is a solution $g_{i+1}(\bar{b})$ for $\psi(\bar{x},g_{i}(u))$ over $g_{i}(A_{i})$ in $cl(P^{\mu}(N))$, that is an o-system over $P^{\mu}(N)$. We use Lemma 15.3 We get $(**)$. 3. (3)

$i_{1}\leq i\leq m$

1. (a)

$A_{i+1}=\langle A_{i}b\rangle$, $b\in R(M)$, $b$ is geometrical independent over $\langle R^{\mu}(M)A_{i}\rangle$.

Since $N$ is $2\times$-rich, we find $g_{i+1}(b)$ in $N$, such that $g_{i+1}(b)$ is geometrically independent over $\langle R^{\mu}(N)g_{i}(A_{i})\rangle$. Then $(**)$ is true. 2. (b)

$A_{i+1}=\langle A_{i}b\rangle$, $[b,d]=e$, $d,e\in A_{i}$, and $b$ is not $cl(P^{\mu}(M))$.

There is some $c\in R(N)$, such that $[c,g_{i}(d)]=g_{i}(e)$. We show $c\notin cl(P^{\mu}(N))$. Then we can extend $g_{i}$ by $g_{i+1}(b)=c$ and get $(**)$.

1. (i)

$d\in cl(P^{\mu}(M))$

Then $d\in C_{i_{1}}$ and $b,e\notin cl(P^{\mu}(M)$.Then $g_{i}(d)\in g_{i}(C_{i_{1}})\subseteq cl(P^{\mu}(N))$ and $g_{i}(e)\notin cl(P^{\mu}(N)$. Hence $c\notin cl(P^{\mu}(N))$. 2. (ii)

$d\notin cl(P^{\mu}(M)$ and $e\notin cl(P^{\mu}(M),d)$

Then we have the same geometrically situation for $g_{i}(d)$ and $g_{i}(e)$ in $N$. Hence $c\in cl(P^{\mu}(N))$ is impossible. 3. (iii)

$d\notin cl(P^{\mu}(M)$ and $e\in cl(P^{\mu}(M),d)$

Then $b\in cl(P^{\mu}(M),d)$. Hence $g_{i}(d)\notin cl(P^{\mu}(N))$ and $g_{i}(e)\in cl(P^{\mu}(N),g_{i}(d))$ and therefore $c\in cl(P^{\mu}(N),g_{i}(d))$. $c\in cl(P^{\mu}(N))$ would imply $g_{i}(d)\in cl(P^{\mu}(N))$, a contradiction. 3. (c)

$A_{i+1}$ is a prealgebraic minimal strong extension of $A_{i}$, generated by an o-system $\bar{b}$ that is an o-system over $P^{\mu}(M)$: $M\models\psi(\bar{b},\bar{c})$, where $\psi$ corresponds to a code formula and $\bar{c}$ is the canonical base algebra, geometrically independent from $cl(P^{\mu}(M))$.

Then $g_{i}(\bar{c})$ and all solutions $\bar{d}$ of $\psi(\bar{x},g_{i}(\bar{c})$ are geometrically independent from $P^{\mu}(N)$. Hence $\bar{d}$ is an o-system over $cl(P^{\mu}(N))$. (**) is true.

$\Box$

Corollary 15.9**.**

Let $M$ be a $2\times$rich $L^{\mu}$-structure. The code formulas $\varphi_{\alpha}(\bar{x},b)$ with $b$ in $P^{\mu}(M)^{\rm eq}$ are strongly minimal.

Proof.

By Lemma 15.3 there are only finitely many solutions in $P^{\mu}(M)$. Let $B\leq M$ be a strong subspace of $P^{\mu}(M)$ such that $b\in{\rm dcl}^{\rm eq}(B)$ and all these solutions in $P^{\mu}(M)$ are in $B$. We show that any two solutions $\bar{a}$, $\bar{c}$ that are not in $B$ have the same $L^{\mu}$-type over $B$. Let $\bar{a}$ and $\bar{c}$ be such solutions of $\varphi_{\alpha}(\bar{x},b)$ not in $B$. By (C3) they are generic over $B$. Then $\langle Ba\rangle$ and $\langle Bc\rangle$ with $f(B)=B$ and $f(a)=c$ fulfil the conditions in Theorem 15.6. Hence ${\rm tp}_{L^{\mu}}(a/B)={\rm tp}_{L^{\mu}}(c/B)$. ∎

Lemma 15.10**.**

*For every good code $\alpha$ there is a $L^{\mu}$-sentence $\chi_{\alpha}$ such that for all $L^{\mu}$-structures $M$ where $M\restriction L\vDash T_{3}$, $P^{\mu}(M)\leq M$, and $P^{\mu}(M)\in\mathcal{K}^{\mu}$ we have :

$M\vDash\chi_{\alpha}$ if and only if every minimal prealgebraic extension of $P^{\mu}(M)$ given by $\varphi_{\alpha}(\bar{x},b)$ with $b\in P^{\mu}(M)^{\rm eq}$ is not in $\mathcal{K}^{\mu}$.*

Proof.

We replace the formula for the code $\alpha$ by the corresponding $L$-formula and denote it by $\varphi_{\alpha}(\bar{x},\bar{y})$ Let $\bar{a}$ be a solution of $\varphi_{\alpha}(\bar{x},\bar{b})$ not in $P^{\mu}(M)$. By C(3) $\bar{a}$ is a generic solution. If $\langle P^{\mu}(M)\bar{a}\rangle$ is not in $\mathcal{K}^{\mu}$, then we have the cases a) or b) of Corollary 13.4. In case a) $\langle P^{\mu}(M)\bar{a}\rangle$ contains a difference sequence for $\varphi_{\alpha}(\bar{x},\bar{b})$ of length $\mu(\alpha)+1$ for $\alpha$. In case b) there is a difference sequence of length $\mu(\beta)+1$ for a good code $\beta$, which contains a subsequence $\ldots,\bar{e_{i}},\ldots$ of length $\mu^{*}(\beta)$ such that $\bar{e_{i}}$ is \mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{w}-generic over $\langle\bar{e_{0}},\ldots,\bar{e_{i-1}}P^{\mu}(M)\rangle$. Hence $\mu^{*}(\beta)n_{\beta}\leq n_{\alpha}$ in this case. Since $\mu^{*}$ is finite-to-one, only a finite set $C_{\alpha}$ of codes $\beta$ can occur. Let $C^{\prime}_{\alpha}=C_{\alpha}\cup\{\alpha\}$. Then $P^{\mu}(M)$ has no prealgebraic minimal extensions in $\mathcal{K}^{\mu}$ given by $\alpha$ if and only if

[TABLE]

[TABLE]

W.l.o.g. $\varphi_{\alpha}(\bar{x},\bar{y})$ is in $L$. Note that the formula after $\exists^{\infty}\bar{x}$ is in $L$ and $\varphi_{\alpha}(\bar{x},\bar{b})$ is strongly minimal in $M\restriction L$. In this way we express ”for $\bar{x}$ generic over $P^{\mu}(M)$”, since by Lemma 15.3 $\varphi_{\alpha}(\bar{x},\bar{b})$ has only finitely many solutions in $P^{\mu}(M)$. ∎

16. Axiomatization of $T^{\mu}_{3}$

By Corollary 15.7 $T^{\mu}_{3}$ is a complete theory. Using the Amalgamation Theorem 14.2 we get a countable $\mathcal{K}^{\mu}$-rich subspace $D\leq\mathfrak{C}$ . Let $P^{\mu}(\mathfrak{C})$ be $D$. Then $(\mathfrak{C},P^{\mu}(\mathfrak{C}^{\mu}))$ is a $2\times$-rich $L^{\mu}$-structure. We call it our standard model. The following is true in $(\mathfrak{C},P^{\mu}(\mathfrak{C}^{\mu}))$. It can be expressed in $L^{\mu}$. Let $M$ be a $L^{\mu}$-structure:

[TABLE]

These sets of axioms are elementary. For $T^{\mu}_{3}\>1)$ this is clear. There is no problem to express $T^{\mu}_{3}\>2)$. For $T^{\mu}_{3}\>3)$ note that $P^{\mu}(M)$ is strong since it is closed under ${\rm acl}^{L}$. The absense of difference sequences for $\varphi_{\alpha}(\bar{x},\bar{y})$ of length $\mu(\alpha)+1$ in $P^{\mu}(M)$ can be expressed by Theorem 12.16. For $T^{\mu}_{3}\>4)$ we use Lemma 15.10. Finally for all axioms we find $L^{\mu}$-formulas.

It is clear that $2\times$-rich $L^{\mu}$-structures satisfy $T^{\mu}_{3}\>1)$, $T^{\mu}_{3}\>2)$ and $T^{\mu}_{3}\>3)$. That they also satisfy $T^{\mu}_{3}\>4)$ is part of the following theorem:

Theorem 16.1**.**

An $L^{\mu}$-structure $M$ that satisfies $T^{\mu}_{3}\>1)$, $T^{\mu}_{3}\>2)$ and $T^{\mu}_{3}\>3)$ is $2\times$rich if and only if it is an $\omega$-saturated model of $T^{\mu}_{3}$.

Proof.

First assume that $M=(M\restriction L,P^{\mu}(M))$ is an $\omega$-saturated model of $T^{\mu}_{3}$. As a saturated model of $T_{3}$ $M\restriction L$ is rich in the sense of $L$. We show that $P^{\mu}(M)$ is $\mathcal{K}^{\mu}$-rich. Let $B\subseteq A$ be in $\mathcal{K}^{\mu}$. Assume $B\leq P^{\mu}(M)$. W.l.o.g. $A$ is a minimal strong extension of $B$. There are three cases:

1. i)

If $A=\langle Ba\rangle$ and $a$ is algebraic over $B$, then $A$ is in $P^{\mu}(M)$ by $T^{\mu}_{3}\>2)$. 2. ii)

Let $A$ be a minimal prealgebraic extension of $B$: $A=\langle B\bar{a}\rangle$ where $\bar{a}$ is the generic solution of some code formula $\varphi_{\alpha}(\bar{x},b)$ where $b$ is in ${\rm dcl}^{\rm eq}(B)$. Let $B\leq D\leq P^{\mu}(M)$ in $\mathcal{H}_{3}$. By Theorem 14.2 an amalgam of $D$ and $A$ over $B$ exists in $\mathcal{K}^{\mu}$ and can be embedded into $M$ over $D$ by the properties of $T_{3}$. If we consider all possible $D$ and none of these amalgams can be embedded into $P^{\mu}(M)$, then we get by $\omega$-saturation some $P^{\mu}(M)\otimes_{B}\langle B\bar{a}\rangle$ in $\mathcal{K}^{\mu}$, a contradiction to axiom $T^{\mu}_{3}\>4)$. 3. iii)

$A$ is a minimal transcendental extension. Then Axiom $T^{\mu}\>2)$ ensures the assertion.

Now let $M$ be a $2\times$rich $L^{\mu}$-structure. $M$ satisfies $T^{\mu}\>1)-T^{\mu}\>3)$. We show $T^{\mu}\>4)$. Let $\varphi_{\alpha}(\bar{x},b)$ be a good code formula. Choose a strong subspace $B$ in $P^{\mu}(M)$ such that $b\in{\rm dcl}^{\rm eq}(B)$. Assume there is a solution $\bar{a}$ of $\varphi_{\alpha}(\bar{x},b)$ generic over $P^{\mu}(M)$ such that $\langle P^{\mu}(M),\bar{a}\rangle$ is in $K^{\mu}$. Since $M$ is $\mathcal{K}^{\mu}$-rich there is a strong copy $A_{0}\supseteq B$ of $\langle B\bar{a}\rangle$ over $B$ in $P^{\mu}(M)$. In the next step we get a copy $A_{1}$ of $\langle A_{0}\bar{a}\rangle$ over $A_{0}$ inside $P^{\mu}(M)$. We can continue this process as long as we want and get a contradiction to the fact that $P^{\mu}(M)$ is in $\mathcal{K}^{\mu}$.

By Corollary 14.4 there exists a $2\times$rich $L^{\mu}$-structure. Hence $T^{\mu}_{3}$ is consistent and we have an $\omega$-saturated model $N$ of $T^{\mu}_{3}$. As shown above $N$ is a $2\times$rich $L^{\mu}$-structure. By Theorem 15.6 $M$ and $N$ are $L_{\infty,\omega}^{\mu}$-equivalent. Hence $M$ is an $\omega$-saturated model of $T^{\mu}_{3}$. ∎

Corollary 16.2**.**

The deductive closure of $T^{\mu}\>1)$ – $T^{\mu}\>4)$ is the complete theory $T^{\mu}_{3}$.

Proof.

This follows from Theorem 16.1 and Corollary 15.7. ∎

Let $\mathfrak{C}^{\mu}$ be the monster model of $T^{\mu}_{3}$ where we work in.

Lemma 16.3**.**

Let $M\preceq\mathfrak{C}^{\mu}$ be a model of $T^{\mu}_{3}$.

1. i)

$R^{\mu}(\mathfrak{C}^{\mu})$* and $R(M)$ are geometrically independent over $R^{\mu}(M)$.* 2. ii)

In $P^{\mu}(\mathfrak{C}^{\mu})$   ${\rm cl}(X)$ is part of ${\rm acl}^{\mu}(X)$. 3. iii)

For $i=1,2$ $R_{i}^{\mu}(x)=R^{\mu}(x)\cap U_{i}(x)$ is strongly minimal. Hence $R^{\mu}(x)$ has Morley rank 1 and Morley degree 2. 4. iv)

$P^{\mu}(x)$* has Morley rank 3.*

Proof.

i) Let $\bar{a}$ in $R(M)$ be geometrically dependent over $R^{\mu}(M)$. Then “$\bar{a}$ is geometrically independent over $R^{\mu}$” is part of the $L^{\mu}$-type of $\bar{a}$. Since ${\rm tp}^{M}(\bar{a})={\rm tp}^{\mathfrak{C}^{\mu}}(\bar{a})$ it follows the assertion.

ii) W.l.o.g. we assume $\langle B\rangle\leq P^{\mu}(\mathfrak{C}^{\mu})$, $B\subseteq R^{\mu}(\mathfrak{C}^{\mu})$ and $a\in{\rm cl}(B)\cap P^{\mu}(\mathfrak{C}^{\mu})$. Then $A=CSS(\langle Ba\rangle)\leq P^{\mu}(\mathfrak{C}^{\mu})$. By C(4) there is a geometrical construction from $B$ to $A$, that has only algebraic and prealgebraic steps. By Lemma 15.3 $A\subseteq{\rm acl}^{\mu}(B)$.

iii) To show the strong minimality of $R^{\mu}_{i}(x)$, we consider again some $\omega$ saturated $M\preceq\mathfrak{C}^{\mu}\vDash T^{\mu}$ and $a,c\in R^{\mu}_{i}(\mathfrak{C}^{\mu})\setminus M$. By ii) $a$ and $c$ are not in ${\rm cl}(R^{\mu}(M))\cap P^{\mu}(\mathfrak{C})$. By i) they are both not in ${\rm cl}(M)$. By Lemma 15.5 every finite subspace of $M$ is contained in some $A\leq M$ that satisfies $(*)$. If we define $f={\rm id}$ on $A$ and $f(a)=c$, then $\langle Aa\rangle$ and $f$ satisfy the conditions of Theorem 15.6. Hence ${\rm tp}_{L^{\mu}}(A,a)={\rm tp}_{L^{\mu}}(A,c)$ and therefore ${\rm tp}_{L^{\mu}}(a/M)={\rm tp}_{L^{\mu}}(c/M)$ as desired.

iv) Let $a,b$ be two geometrically independnet elements of $R^{\mu}_{1}(\mathfrak{C}^{\mu})$. The function $[[a,b],x]$ gives a defiable bijection between $R^{\mu}_{1}(\mathfrak{C}^{\mu})$ and $P^{\mu}(\mathfrak{C}^{\mu})_{3}=\langle R^{\mu}(\mathfrak{C}^{\mu})\rangle_{3}$. Hence $P^{\mu}(\mathfrak{C}^{\mu})_{3}$ is strongly minimal. Hence $P^{\mu}(x)$ has finite Morley rank 3. ∎

Now we give another definition for the property (*) in Theorem 15.6.

Lemma 16.4**.**

1. (1)

A finte strong substructure $A\in\mathcal{H}_{3}$ of $\mathfrak{C}^{\mu}$ satisfies () iff $cl(P^{\mu}(\mathfrak{C}))\cap A=cl(R^{\mu}(\mathfrak{C}^{\mu})\cap A)\cap A$.* 2. (2)

We use (1) to define () for $U\leq\mathfrak{C}^{\mu}$ in $\mathcal{H}_{3}$. Then () for $U$ is equivalent to: For every $A\subseteq U$ there is some $A\subseteq C\leq\mathfrak{C}^{\mu}$, that fullfils ().* 3. (3)

If $U\leq\mathfrak{C}^{\mu}$ and $V\leq\mathfrak{C}^{\mu}$ are isomorphic strong substructures in $\mathcal{H}_{3}$, that satisfy (), then they have the same type.*

Theorem 16.5**.**

$T_{3}^{\mu}$* is $\omega$-stable.*

Proof.

Let $M$ be a countable elementary submodel of $\mathfrak{C}^{\mu}$. We show that there are only countably many types ${\rm tp}(\bar{a}/M)$ where $\bar{a}$ is a finite tuple in $\mathfrak{C}^{\mu}$ . W.l.o.g. we can restrict us to $\bar{a}\subseteq R(\mathfrak{C}^{\mu})$. Furthermore we will consider finite subspaces $\bar{a}\subseteq A\subseteq R(\mathfrak{C})$ with certain properties only. For a given $\bar{a}\subseteq R(\mathfrak{C}^{\mu})$ it is easy to find a set XYZW of geometrically independent elements (short geo. basis) such that the following is true:

1. (0)

$\bar{a}\subseteq{\rm cl}_{d}(XYZW)$ 2. (1)

$X\subseteq R^{\mu}(M)$ 3. (2)

$Y\subseteq R(M)$ is geometrically independent over $R^{\mu}(M)$. 4. (3)

$Z\subseteq R^{\mu}(\mathfrak{C}^{\mu})$ (short $R^{\mu}$) is geometrically independent over $R^{\mu}(M)$. 5. (4)

$W$ is geometrically independent over $\langle R(M),R^{\mu}\rangle$.

By Lemma 16.3 i) $Y$ is geometrically independent over $R^{\mu}$ and $Z$ over $M$. Now we choose any $A$ such that $XYZW\subseteq A\subseteq{\rm cl}(XYZW)$, $\bar{a}\subseteq A$ and $A\leq\mathfrak{C}^{\mu}$. $A$ is obtained by a sequence of minimal strong extensions over $A\cap M$. We use (C2), (C3), and (C4). Since $d(A/M)=d(A/A\cap M)$, Lemma 12.1 implies that $\langle MA\rangle$ is strong in $\mathfrak{C}^{\mu}$ and it is obtained by the same sequence over $M$. Note that $\langle MA\rangle$ satisfies (*). If $A\cap M=A^{\prime}\cap M$, $A$ and $A^{\prime}$ are isomophic over this intersection, and $A^{\prime}\leq\mathfrak{C}^{\mu}$, then they have the same type over $M$ by Lemma 16.4. Hence there are only contably many types over $M$.

∎

17. A Lie algebra of finite Morley rank

Let $T_{i}$ ($i=0,1$) be complete $L_{i}$-theories. Let $\Delta$ be an interpretation of the theory $T_{0}$ in the theory $T_{1}$. In [5] is defined that $\Delta$ is an interpretation of $T_{0}$ in $T_{1}$ without new information, if for every $M\vDash T_{1}$ every subset $X$ of $\Delta(M)^{n}$ defined in $M$ by a $L_{1}$-formula without parameters is definable by a $L_{0}$-formula without parameters. If $T_{1}$ is stable, then we have the same for formulas with parameters. In [5] the following result of Lascar is published:

Lemma 17.1** (Lascar).**

If $T_{1}$ is stable and $\Delta$ is an interpetation of $T_{0}$ in $T_{1}$ without new information, then for every model $N$ of $T_{0}$ there is some model $M\vDash T_{1}$ such that $\Delta(M)\cong N$.

Definition 17.2**.**

If $M$ is a model of $T_{3}^{\mu}$, then let $\Gamma(M)$ be the $L$-substructure of $M$ with domain $P^{\mu}(M)$. Let $\Gamma(T_{3}^{\mu})$ be the complete $L$-theory of all $\Gamma(M)$ where $M\vDash T_{3}^{\mu}$.

$\Gamma$ defined above is an interpetation. We get:

Theorem 17.3**.**

Every subset of $P^{\mu}(\mathfrak{C}^{\mu})^{n}$ 0-defined in $\mathfrak{C}^{\mu}$ is $L$-0-definable in $\Gamma(\mathfrak{C}^{\mu})$. Hence $\Gamma$ is an interpretation without new information and every model of $\Gamma(T^{\mu})$ has the form $\Gamma(M)$ with $M\vDash T^{\mu}$.

Proof.

Let $A$ and $C$ be finite substructures of $P^{\mu}(\mathfrak{C})$ in $\mathcal{H}_{3}$. It is sufficient to show:

If $A$ and $C$ have the same $L$-type in $\Gamma(\mathfrak{C}^{\mu})$, then they have the same $L^{\mu}$-type in $\mathfrak{C}^{\mu}$.

By the assumption there is an $L$-isomorphism of $A$ onto $C$, that can be extended to an isomorphism $f$ of $CSS(A)$ onto $CSS(C)$ in $\Gamma(\mathfrak{C}^{\mu})$. Since $P^{\mu}(C^{\mu})\leq\mathfrak{C}^{\mu}$ the self-sufficient closure in $\Gamma(\mathfrak{C}^{\mu})$ is the same as in $\mathfrak{C}^{\mu}$. By Corollary 15.8 $A$ and $C$ have the same type in $\mathfrak{C}^{\mu}$. ∎

Since $T^{\mu}_{3}$ is $\omega$-stable we get:

Corollary 17.4**.**

Every subset of $P^{\mu}(\mathfrak{C}^{\mu})^{n}$ defined in $\mathfrak{C}^{\mu}$ with parameters is definable in $\Gamma(\mathfrak{C}^{\mu})$.

Theorem 17.5**.**

For every suitable function $\mu$ $\Gamma(T^{\mu}_{3})$ is uncountably categorical. $R_{1}(x)$ and $R_{2}(x)$ are a strongly minimal formulas in this theory. The pregeometry of $R$ is given by ${\rm acl}={\rm cl}$. For models $N$ of $\Gamma(T^{\mu}_{3})$ we have $N=\langle R(N)\rangle$. The Morley rank of $\Gamma(T^{\mu}_{3})$ is 3. The geometry of the theory is not locally modular.

Proof.

$R^{\mu}_{i}(x)$ for $i=1,2$ are strongly minimal for $T^{\mu}$ by Lemma 16.3 iii). Hence $R_{i}(x)$ is strongly minimal in $\Gamma(T^{\mu})$. Since $\Gamma(M)=\langle R_{1}(\Gamma(M))\rangle$ we have that $\Gamma(T^{\mu})$ is uncountably categorical. Since ${\rm cl}$ contains ${\rm acl}$ we get ${\rm acl}={\rm cl}$ by Lemma 15.3. ∎

Corollary 17.6**.**

$\Gamma(T_{3}^{\mu})$* is CM-trivial.*

It is not possible to interpret an infinite field in $\Gamma(T_{3}^{\mu})$.

Proof.

We follow the proof of the CM-triviality for $T_{3}$ inside $P^{\mu}(\mathfrak{C})$. Especially we can use modified versions of the Lemmas 8.3, 8.8 and 8.9. ∎

18. New uncoutably categorical groups

We assume that $M$ is a graded Lie algebra over the field $\mathbb{F}(p)$, where p is a prime greater than 3 and $\mathbb{F}(p)$ is the field with p elements. The Baker-Campbell-Hausdorff-formula provides us a group multiplication on the domain of M:

[TABLE]

Normally we have an infinite sum and characteristic 0. Since $M$ is 3-nilpotent the sum is finite. The usual proof works. In $\mathbb{F}(p)$ all coefficients in the considered finite series exists, since $3<p$. Let $\mathbb{G}(M)$ be the group defined above. [math] is the unit in this group. The inverse element of $x$ is $-x$.

We compute the group commutator:

[TABLE]

[TABLE]

[TABLE]

Hence

[TABLE]

Note that $rx$ is $x\circ x\circ\ldots\circ x$ with $r-1$ use of $\circ$. Then we get conversely

[TABLE]

Next we define $x+y$ using only the group-multiplication. As shown above we can use the Lie multiplication and $rx$.

[TABLE]

If $M$ is a free 3-nilpotent Lie algebra over $\mathbb{F}(p)$, then $\mathbb{G}(M)$ is free in the variety of 3-nilpotent groups of exponent p.

If $M$ is generated by $M_{1}$, then $\mathbb{G}(M)$ is generated by $M_{1}$.

Let $L^{-}$ be the reduction of $L$, where the predicates $U_{i}$ and the projections are canceled. Let $M^{-}$ be the corresponding $L^{-}$-reduct of $M$. $\circ$ is definable in $M^{-}$. Conversely the $L^{-}$-reduct $M^{-}$ of $M$ is definable using $\circ$ only, as shown above. Hence we can show:

Lemma 18.1**.**

The elementary theories of $\mathbb{G}(\Gamma(\mathfrak{C}^{\mu}))$ and $(\Gamma(\mathfrak{C}^{\mu})^{-}$ are binterpretable.

In fact the group $\mathbb{G}(M)$ can be considered as the $L^{-}$-reduct of $M$.

Theorem 18.2**.**

The elementary theory of $\mathbb{G}(\Gamma(\mathfrak{C}^{\mu}))$ is uncountably categorical of Morley rank 3. It is not one-based but CM-trivial. $Z_{1}(\mathbb{G}(\Gamma(\mathfrak{C}^{\mu})))$ is strongly minimal.

Proof.

We have the results for $\Gamma(T_{3}^{\mu})$ in Theorem 17.5 and Corollary 17.6. Then it follows for $\Gamma(T_{3}^{\mu})^{-}$. For CM-trivialty we can use H.Nübling’s result in [16]: Reducts of CM-trivial stable theories of finite Lascar rank are CM-trivial. By biinterpretability in Lemma 18.1 we get the desired properties for $\mathbb{G}(\Gamma(T_{3}^{\mu}))$. For CM-triviality we can use [5]. ∎

Theorem 18.3**.**

The elementary theory of $\mathbb{G}(\Gamma(\mathfrak{C}^{\mu}))$ does not allow the interpretation of an infinite field.

Proof.

Such an interpretation would imply an interpretation of that field in $\Gamma(\mathfrak{C}^{\mu})$, a contradiction. ∎