This paper extends the concept of lexicographic products to infinite iterations, characterizes dense substructures in such products, and constructs a rigid elementarily indivisible structure with unique properties.
Contribution
It introduces a framework for infinite lexicographic products, characterizes their dense substructures, and constructs a new rigid, elementarily indivisible structure.
Findings
01
Any countable product of countable transitive homogeneous structures has a unique dense substructure.
02
The dense substructure is transitive, homogeneous, and elementarily embeds into the product.
03
Constructs a rigid elementarily indivisible structure.
Abstract
We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.
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Department of Mathematics,
Ben Gurion University of the Negev
P.O.B. 653,
Be’er Sheva 8410501, Israel.
Abstract.
We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.
The work in this paper is part of the author’s Ph.D. studies at the Department of Mathematics, Ben-Gurion University of the Negev under the supervision of Assaf Hasson.
The author was Partially supported by ISF Grant 181/60 and the Hillel Gauchman scholarship.
1. Introduction
Much of mathematics in general deals with the construction of new mathematical structures using existing ones as building blocks. Examples of such constructions are pervasive throughout mathematics. In algebra, there are constructions such as direct products, wreath products, and tensor products. In topology, there is the product topology, a.k.a. the product space. In graph theory and combinatorics, there are numerous notions of a product of two given graphs, such as the Cartesian product, the tensor product, and the lexicographic product. Even in naïve set theory, the Cartesian product of sets plays a primary role, and, in axiomatic set theory, the existence of a Cartesian product of infinitely many sets is a source of long standing debates.
The study of properties of the new structures relative to their building blocks is central in each of the mathematical branches mentioned above. Examples span from classification of subgroups of a direct product of groups, through classical results in topology regarding preservation of separation axioms under products, and lack thereof, to
the calculation of the chromatic number of a product of two graphs by means of arithmetic on their chromatic numbers. Indeed, the author has yet to find a major branch of mathematics in which this is not the case.
In [Mei16], the author studied a construction of the same nature, in the context of first-order relational structures, as defined below.
Definition 1.1**.**
Let M, {Na}a∈M be structures in a relational language, L. Let M,{Na}a∈M be their universes, respectively.
The generalized lexicographic productM[Na]a∈M is the L-structure whose universe is ⋃a∈M{a}×Na where for every
n-ary relation R∈L we set RM[Na]a∈M to be
[TABLE]
In case N=Na for all a∈M, this is abbreviated by M[N] and this construction generalizes the lexicographic order and the lexicographic product of graphs. More generally, in a binary language, M[N] coincides with a classical construction denoted by the same notation. (e.g. [Che98, Lac87].)
Let M[Na]a∈Ms be M[Na]a∈M
expanded by a binary relation s∈/L interpreted as
\Set{\big{(}(a,b_{1}),(a,b_{2})\big{)}}{a\in\mathcal{M}\text{\ \ and\ \ }b_{1},b_{2}\in\mathcal{N}_{a}}.
Let Nas is Na expanded by a binary relation s interpreted as (Na)2 for all a∈M. Then
M[Na]a∈Ms=M[Nas]a∈M. For this reason, we identify the two constructions and denote both by M[Nas]a∈M.
In [Mei16], the author proved several results demonstrating the good behavior of the lexicographic product with regards to elementary equivalence, elementary embeddings, quantifier elimination, etc. (See Subsection 2.2 for a reveiw of some of these results.)
Consequenty, many model-theoretic properties such as simplicity, stability, NIP, etc. are preserved under lexicographic products.
The study of lexicographic products was motivated by several question regarding elementarily indivisible structures, as defined below. A first-order relational structure is indivisible if for every colouring of its universe in two colours, there is a monochromatic substructure isomorphic to it. Additionally, it is elementarily indivisible if the monochromatic substructure can be chosen to be elementary. Indivisibility of relational first-order structures is a well studied notion
in Ramsey theory
(e.g., [KR86], [EZS94], [EZS93], and [Fra00, Appendix A]). In [HKO11] by Hasson, Kojman and Onshuus asked three question concerning elementarily indivisible structure, two of them answered in [Mei16]. The motivation behind the research presented in this paper is the third and final question in [HKO11]:
Is there a rigid elementarily indivisible structure?
Here, by rigid, we mean a structure whose automorphism group is trivial.**
In this paper, we take the lexicographic product a step further, to infinity (and beyond),
by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times.
We then concentrate on the case of countably many iterations, in which we define the notion of a dense substructure; we prove that for any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous. In addition, we show that such a dense substructure not only elementarily embeds into the product, but also there is an Lω1,ω-elementary embedding of it into the product.
As an application, we show that a dense substructure in a countable lexicographic products of countable homogeneous indivisible structures is elementarily indivisible and answer 1.2.
We conclude by defining a strengthening of the elementary indivisibility to Lω1,ω and show that every Lω1,ω-elementarily indivisible structure is transitive.
Acknowledgements
The author is grateful to Assaf Hasson for presenting the question which motivated this paper, as well as for the fruitful discussions and the warm support along the way.
2. Preliminaries
In this section we summarize some of the context for our results, including a few basic definitions from model theory, mainly concentrating on countable infinitary logic, i.e., Lω1,ω and its relation to homogeneity, as well as several results from [Mei16] that this paper generalizes.
Unless otherwise specified, we do not distinguish between a structure M and its universe (or underlying set). Throughout this paper all languages are relational, so there is no distinction between subsets and substructures of a given structure. The notation for both is B⊆M. We denote the cardinality of a structure M by ∣M∣.
2.1. Homogeneity and infinitary logic
Definition 2.1**.**
If M and N are L-structures and B⊆M, then f:B→N is a
partial isomorphism if
M⊨φ(bˉ)⟺N⊨φ(f(bˉ))
for all quantifier-free (or equivalently, atomic) L-formulas φ and all finite sequences bˉ from B.
Definition 2.2**.**
A structure M is homogeneous if whenever A⊂M with ∣A∣<∣M∣ and f:A→M is a partial
isomorphism, there is an automorphism σ∈Aut(M) such that σ↾A=f.
Definition 2.3**.**
An Lω1,ω-theory T admits quantifier elimination (QE) if for every Lω1,ω-formula ϕ there is a quantifier-free Lω1,ω-formula ψ such that
T⊨ϕ↔ψ. An L-structure M admits Lω1,ω-QE if its Lω1,ω-theory admits QE.
Definition 2.4**.**
Let L be a first-order language and let vˉ=v1,…,vn.
(1)
An L-diagram in variables vˉ is a (perhaps partial) type p consisting of only atomic and negated atomic L-formulas.
2. (2)
An L-diagram in variables vˉ is complete if for every k-ary R∈L and every 1≤i1,…,ik≤n, either R(vi1,…,vik)∈p or ¬R(vi1,…,vik)∈p.
3. (3)
An L-diagram in variables vˉ is T-consistent for T, where T is either an L-theory or an Lω1,ω-theory if
T⊨¬∃vˉ⋀ϕ∈pϕ(vˉ).
Lemma 2.5**.**
Let L be a first-order language. If M is an L-structure of size κ≥ℵ0, then for any complete L-diagram p in variables vˉ, there is some L-diagram q of size κ in variables vˉ such that
aˉ⊨p⟺aˉ⊨q for all aˉ∈M.
Proof.
Assume not. We construct, by induction, a sequence of pairwise distinct tuples {aˉα}α<κ+⊆M and formulas {ϕα}α<κ+⊆p such that aˉβ⊨ϕβ and aˉβ⊨ϕα for all α<β<κ+. This will contradict ∣M∣=κ.
•
There is some ϕ0∈p and aˉ0∈M such that aˉ0⊨ϕ0.
•
Assume aˉα and ϕα were defined for all α<β<κ+. Since β<κ+, there is some ϕβ∈p and aˉβ∈M such that aˉβ⊨ϕα for all α<β<κ+ but aˉβ⊨ϕβ.
By the construction, aˉα⊨ϕα and aˉβ⊨ϕα for all α<β<κ+. so aˉα=aˉβ.
∎
Lemma 2.6**.**
Let L be a first-order language, let M be a countable L-structure and let ϕ(vˉ) be a quantifier-free Lω1,ω-formula.
If ϕ is quantifier-free or M is homogeneous, then ϕ has a disjunctive normal form, i.e., a formula of the form ⋁j∈J⋀i∈Ijθi(vˉ) such that J and Ij are countable for all j∈J and θi is atomic or negated atomic for all i∈⋃j∈JIj and
[TABLE]
Proof.
Let Ψ be the set of all complete L-diagrams realized in M. By Lemma 2.5, for any p∈Ψ, there is some countable qp such that aˉ⊨p⟺aˉ⊨qp for all aˉ∈M. Let Ψ′:={qp}p∈Ψ. Then for any aˉ∈M, there is some q∈Ψ′ such that aˉ⊨q.
•
If ϕ is quantifier-free, then for any complete L-diagram p, either p⊢ϕ or p⊢¬ϕ.
•
If M is homogeneous, then for any two tuples aˉ,bˉ satisfying the same complete L-diagram, there is a partial isomorphism f:aˉ→bˉ, which in turn extends to an automorphism of M. So aˉ,bˉ satisfy the same Lω1,ω-formulas.
In conclusion, if either ϕ is quantifier free or M is homogeneous, then for any complete L-diagram p∈Ψ, we have that either aˉ⊨p⟹M⊨ϕ(aˉ) for all aˉ∈M, or aˉ⊨p⟹M⊨ϕ(aˉ) for all aˉ∈M.
So for any q∈Ψ′ we have that either aˉ⊨q⟹M⊨ϕ(aˉ) for all aˉ∈M, or aˉ⊨q⟹M⊨ϕ(aˉ) for all aˉ∈M. Let
[TABLE]
So Ψ′=Ψ1∪Ψ2, therefore,
M⊨ϕ(aˉ)⟺⋁q∈Ψ1aˉ⊨q
for all aˉ∈M.
Since M is countable, so is Ψ1 as a set of realized diagrams. Since every q is a countable L-diagram, we can write aˉ⊨q⟺M⊨⋀θ∈qθ(aˉ) for all aˉ∈M and the right hand side is a countable disjunction of atomic and negated atomic formulas. In conclusion
[TABLE]
and Ψ1,q are countable.
∎
Lemma 2.7**.**
Let M be a countable structure. Then M is homogeneous if and only if M admits Lω1,ω-QE, which, in turn, implies that every embedding between isomorphic copies of M is elementary.
For ⇐: Assume M admits Lω1,ω-QE, let f:aˉ→bˉ be a finite partial isomorphism and c∈M. We want to find some d∈M such that f∪⟨c,d⟩ is a partial isomorphism.
Let p(vˉ,x) be the complete L-diagram realized by aˉ,c. By Lemma 2.5, there is some countable L-diagram q equivalent to p in M. It suffices to show that M⊨∃x⋀θ∈qθ(bˉ,x). Indeed, M⊨∃x⋀θ∈qθ(aˉ,x) and by Lω1,ω-QE, there is some quantifier-free Lω1,ω-formula φ(vˉ) such that M⊨∀vˉ(∃x⋀θ∈qθ(vˉ,x)↔φ(vˉ)). So M⊨φ(aˉ) and, since φ is quantifier free, M⊨φ(bˉ) so M⊨∃x⋀θ∈qθ(bˉ,x).
∎
Let L be a relational language, let s∈/L be a binary relation symbol and let T1,T2 be L-theories (not necessarily complete).
If T1 and T2 both admit QE and T1 has a transitive model then
there is an L∪{s}-theory T (not necessarily complete) admitting QE, such that M[Na]a∈Ms⊨T whenever M⊨T1 and {Na}a∈M⊨T2.
In particular, if M and N are L-structures both admitting QE and M is transitive
then M[N]s admits QE.
Let M,{Na}a∈M;M′,{Na′}a∈M′ be structures in a relational language, L, such that Th(M) has transitive models. If M≺M′ and Na≺Na′ for all a∈M then M[Na]a∈Ms≺M′[Na′]a∈M′.s
Proof.
Consider the Morleyzations M,{Na}a∈M;M′,{Na′}a∈M′ as defined in in [Mei16, Notation 2.19]. By definition of the Morleyzation, there is an L-theory T eliminating quantifiers, such that all Morleyzations of L-structures model T. Since M≺M′ and Na≺Na′ for all a∈M, it follows that M≺M′ and Na≺Na′ for all a∈M. By 2.8, M′[Na′]a∈M′s and M[Na]a∈Ms both model an L∪{s}-theory which eliminates quantifiers, so the canonical embedding M[Na]a∈Ms↪M′[Na′]a∈M′s is elementary.
∎
Proposition 2.11**.**
(1)
If M and N are transitive, then M[Ns] is transitive.
2. (2)
If M and N are κ-homogeneous for some cardinal κ, then M[Ns] is κ-homogeneous.
Proof.
(1)
Let a,b∈M,c,d∈N and f∈Aut(M),g∈Aut(N) such that f(a)=b,g(c)=d. Then, for F∈Aut(M[Ns]) defined by F((x,y)):=(f(x),g(y)), we have F((a,b))=(c,d).
2. (2)
Let λ<κ and let ⟨(ai,bi)⟩i<λ+1,⟨(ci,di)⟩i<λ be sequences of elements in M[Ns] such that
[TABLE]
We need to find some (cλ+1,dλ+1) such that
[TABLE]
By κ-homogeneity of M and N there are cλ+1∈M and dλ+1∈N such that
tpMqf(⟨ai⟩i<λ+1)=tpMqf(⟨ci⟩i<λ+1)
and
tpNqf(⟨bi⟩i<λ+1)=tpNqf(⟨di⟩i<λ+1).
By definition of M[Ns], we are done.
∎
3. Finite tree products
We can iterate the product defined in Definition 1.1 any finite number of times, and this product is, in fact, associative: using the bijection (a,(b,c))↦((a,b),c), we get
M[N[P]s2]s1≅(M[N]s1)[P]s2 and
[TABLE]
If I is a structure whose universe is a singleton and I,M⊨∀x¬R(x,…,x) for all R∈L, then
M[I]≅I[M]≅M.
Next, we demonstrate how any finitely iterated product as above is equivalent to a product induced by a tree of finite height.
Consider the example of M[Na[Pb]b∈Nas2]a∈Ms1. If we assume, for simplicity, that all structures in the product are structures on ω as their underlying set. Observe the tree illustrated below.
In this tree, each internal (non-leaf) node of the tree is associated with a structure, S(t). For such a node t, the set of immediate successors of t are indexed by the universe of S(t). Thus, to any node t∈T (except the root) is associated a unique element, e(t), of the structure inhabiting its immediate predecessor.
For every k-tuple of leaves of the tree (a1,…,ak) such that ⋁1≤i<j≤nai=aj we can find some node m in the tree such that m is the meet of a1,…,ak, i.e. m=a1∧⋯∧ak:=max{x}x≤a1,…,ak. Notice that every chain in the tree is discretely-ordered, and thus, m has an immediate successor in the segment [m,ai]:={x}m≤x≤ai; call it Sai(m).
So in the tree products, for every k-ary relation R∈L,
[TABLE]
and we denote
si(a,b)⇔height(∧(a,b))≥i.
Notice that the tree product described above is isomorphic to M[Na[Pb]b∈Nas2]a∈M.s1
In the same sense as above, any finitely iterated product is isomorphic to a product induced by a tree of finite height, defined below.
Definition 3.1**.**
Let ⟨T,<⟩ be a tree of finite height, where:
•
leaf(T) is the set of <-maximal elements in T.
•
succ(t):={s∈T}t<s∧∃x(t<x<s) for t∈T.
•
height(t) is order type of the set {s∈T}s<t.
•
height(T):=maxt∈T(height(t)).
If (Mt)t∈T∖leaf(T) is a family of structures in a relational language L indexed by T, such that each Mt is a structure whose universe is succ(t), then we define the tree product ∏TMt to be the L-structure whose universe is leaf(T) where for every k-ary relation R∈L we set R∏TMt to be
[TABLE]
If ⟨sα⟩1≤α<height(T) is a sequence of pairwise distinct binary relation symbols disjoint from L, let ∏TMts be an expansion of ∏TMt to
L∪{si}1≤i<height(T), where sα is interpreted as
{(a,b)}height(a∧b)≥i.
Remark 3.2**.**
Let M be an L-structure such that M⊨∀x¬R(x,…,x) for every R∈L.
Let T:={r}∪M such that M=succ(r) and let Mr:=M. Then M≅M[I]≅∏TMt.
By finite induction, results from [Mei16] such as 2.8 and Propositions 2.10 and 2.11 easily extends to tree products where height(T) is finite. In the following section, we generalize some of these results to the case where T may be of infinite height.
4. Infinite tree products
In this section, we rigorously defining an infinite iteration process as a product of a tree of structures, not necessarily of finite height.
In Section 5, we concentrate on the case where the tree is countable; beforehand, we define and study some of the basic properties of a product induced by a successor meet tree of any size or height, defined below.
Definition 4.1**.**
A successor meet tree is a partially ordered set ⟨T,<⟩, such that the following hold:
(1)
For all t∈T, the set T<t:={s∈T}s<t is a chain.
2. (2)
For every maximal chain C⊆T and a∈C, if a is not maximal, then a has an immediate successor in C: there is some s∈C such that a<s and for all s′∈C, if a<s′ then s≤s′.
*We denote the immediate successor s of a in C by SC(a).
*
3. (3)
Every a,b∈T have a meetm∈T: there is some m≤a,b such that for all m′∈T, if m′≤a,b then m′≤m.
We denote the meet m of a and b by a∧b.
Notation 4.2**.**
Let ⟨T,<⟩ be a successor meet tree.
•
branch(T)* is the set of maximal <-chains.*
•
leaf(T)* is the set of <-maximal elements in T.*
•
int(T)=T∖leaf(T).
•
succ(t):={s∈T}t<s∧∃x(t<x<s)* for t∈T.*
•
T<t:={s∈T}s<t* for t∈T.*
•
T<A:={s∈T}∃a∈A(s<a)=⋃a∈AT<a* for A⊆T.*
Similarly we define T≤t, T>t, T≥t ; T≤A, T>A, T≥A.
Remark 4.3**.**
If T is a successor meet tree, then so is T≥t for all t∈T, and if A⊂T is a maximal anti-chain, then T≤A is a successor meet tree as well.
Lemma 4.4**.**
Let ⟨T,<⟩ be a successor meet tree.
(1)
If B∈branch(T) and b∈B, then T<b⊂B.
2. (2)
If B,C∈branch(T) such that B=C, then there is some t∈int(T) such that B∩C=T≤t. We denote such t by B∧C.
Moreover, if b∈B∖C and c∈C∖B then B∧C=b∧c.
Proof.
(1)
Otherwise, by maximality of B, there is some c∈B and a∈T<b such that c≰a and a≰c. Therfore, b≰c, so c∈T<b, contradicting T<b being a chain.
2. (2)
By maximality, there are b∈B∖C and c∈C∖B. We claim that B∩C=T≤b∧c.
Indeed, It follows from Item 1 that T≤b∧c⊆B∩C.
To prove T≤b∧c⊇B∩C, if there is some a∈B∩C∖T≤b∧c, then since B∩C is a chain, a>b∧c and therefore b,c∈T<a. By Item 1, B∈/T<c and c∈/T<b, contradicting T<a being a chain.
Finally, b∧c∈int(T) since (b∧c)<b,c.
∎
Definition 4.5**.**
(1)
Let ⟨T,<⟩ be a successor meet tree. If (Mt)t∈int(T) is a family of structures in a relational language L indexed by T, such that each Mt is a structure whose universe is succ(t), then we call ⟨T,<,(Mt)t∈int(T)⟩ a family tree. (abbreviated by ⟨T,Mt⟩)
2. (2)
If ⟨T,Mt⟩ is a family tree, we define the product ∏TMt to be the L-structure whose universe is branch(T) where for every k-ary relation R∈L we set R∏TMt to be
[TABLE]
Example 4.6**.**
Recall that ω<ω is the set of all functions f:n→ω for some natural number n, and ωω is the set of all functions from ω to ω.
For each a∈ωω, let a↾n be the restriction of a to n, which is in ω<ω.
We consider the order on ω<ω induced by inclusion of functions, i.e. for t,s∈ω<ω, we define t≤s if there is some n∈ω such that t=s↾n.
This is indeed a partial order, and, in fact, a successor meet tree.
The following illustrates the order on ω<ω, where the maximal chains in the order are precisely the elements of ωω.
Let (Mt)t∈ω<ω be a family of countable structures in a relational language L. We assume that for all t∈ω<ω, the universe of Mt is succ(t).
The product ∏ω<ωMt is the L-structure whose universe is ωω where for every a1,…,ak∈ωω and every k-ary relation R∈L, let n∈ω be maximal such that a1↾n=⋯=ak↾n=:s. Then
[TABLE]
Example 4.7**.**
Recall that ω∗ is the set of natural numbers endowed with the reverse ordering, i.e. ⋯<∗2<∗1<∗0.
For the purposes of this paper, we identify ⟨ω∗,<⟩ with the set of negative integers, endowed with the standard linear order on the integers, i.e., ω∗={−1,−2,−3,…} and ⋯<−3<−2<−1.
In Example 4.6, we took, as the index set for the family of structures, all initial segments of ωω, which turn out to be ω<ω.
Here we take all initial segments of the set B:={a∈ωω∗}a has finite support.
For every b∈B, an initial segment of b is of the form b↾{n∈ω∗}n<m for some m∈ω∗.
Now, we define S to be all initial segments of B, i.e.:
[TABLE]
We endow S with an order, similar to that of ω<ω:
a≤b⇔a⊑b
where ⊑ is the relation stating a is an initial segment of b.
S with this order is a successor meet tree as well.
The following illustrates the order on S. In this case, the maximal chains coincide with the maximal elements in the order, which are the elements of B.
Let ⟨Ms⟩s∈int(S) be a family of countable structures in a relational language where for all s∈int(S), the universe of Ms is succ(s).
As in Example 4.6, the product ∏SMs is the L-structure whose universe is B where for every a1,…,ak∈B and every k-ary relation R∈L, let n∈ω∗ be maximal such that a1↾n=⋯=ak↾n=:s. Then
[TABLE]
Lemma 4.8**.**
Let T1,T2 be successor meet trees, such that ∣succ(t)∣=ℵ0 for any t∈int(T1)∪int(T2).
(1)
If the order type of all branches in T1 and T2 is ω then T1≅T2.
2. (2)
If the order type of all branches in T1 and T2 is ω∗ then T1≅T2.
3. (3)
If the order type of all branches in T1 and T2 is Z then T1≅T2.
Proof.
In all three cases the proof goes as follows. Let B∈branch(T1),C∈branch(T2).
We construct, by induction a sequence of subsets A0⊆A1⊆A2⊆⋯⊆T1 such that ⋃i<ωAi=T1, and a sequnec of partial isomorphisms fn:An→T2 such that f0⊆f1⊆f2⊆… and for every i<ω and a∈Ai, either succ(a)⊆Ai and succ(fi(a))⊆fi(Ai), or ∣succ(a)∖A∣=∣succ(fi(a))∖fi(A)∣=ℵ0. So ⋃i<ωfi will be an isomorphism.
•
By the assumption, there is an order isomorphism f0:B→C and let A0:=B.
•
Let n<ω and assume fn:An→T2 is a partial isomorphism as in the induction hypothesis. For all a∈An, such that ∣succ(a)∖An∣=ℵ0, let ⟨si(a):i<ω⟩ and ⟨ti(a):i<ω⟩ be enumerations of succ(a) and succ(fn(a)), respectively. Then we define
[TABLE]
Finally, if all branches of T1 are of order type ω, ω∗, or Z, then for any t∈T1, there is some b∈B and n∈N such that t∈succn(b), so ⋃i<ωAi=T1.
∎
Definition 4.9**.**
A family tree isomorphism between family trees ⟨T,Mt⟩ and ⟨U,Nu⟩ is a bijective, order preserving function θ:U→T such that θ↾Nu:Nu→Mθ(u) is an isomorphism of L-structures for all u∈U.
If there is such an isomorphism, then ⟨T,Mt⟩ and ⟨U,Nu⟩ are isomorphic, denoted by ⟨T,Mt⟩≅⟨U,Nu⟩.
Remark 4.10**.**
If ⟨T,Mt⟩ and ⟨U,Nu⟩ are isomorphic, then ∏TMt≅∏UNu.
Definition 4.11**.**
Let S be a successor meet tree and let ⟨TB⟩B∈branch(S) be a family of successor meet trees indexed by the branches of S. Then we define S∗⟨TB⟩B∈branch(S) to be the set int(S)∪⋃B∈branch(S)TB with an order < defined by
[TABLE]
Remark 4.12**.**
If T is a successor meet tree and A⊂T is a maximal anti-chain, then A∩B={sup(B)} for all B∈branch(T≤A) and
[TABLE]
Remark 4.13**.**
For any n∈Z, the set T≤n:={t∈T}height(t)=n is a maximal anti-chain.
Corollary 4.14**.**
height(sup(B))=n* For any B∈branch(T≤n) and*
[TABLE]
Lemma 4.15**.**
Let S be a successor meet tree and let ⟨TB⟩B∈branch(S) be a family of successor meet trees, such that for any B∈branch(S), if B has a maximal element, then TB has a minimal element.
(1)
T:=S∗⟨TB⟩B∈branch(S)* is a successor meet tree.*
2. (2)
Populating T, so that ⟨T,Mt⟩ is a family tree,
so are ⟨S,Ms⟩ and ⟨TB,Mt⟩ for all B∈branchS.
Furthermore, there is an isomorphism
[TABLE]
such that f({D∈∏TMt}D∩TB=∅)={B}×∏TBMt for all B∈branch(S). Moreover, if D∩TB=∅, then f(D)=(B,D∩TB).
Proof.
(1)
Exercise.
2. (2)
We will define an isomorphism g:∏SMs[∏TBMt]B∈branchS≅∏TMt and the wanted f will be g−1.
Let g be defined by (B,C)↦(B∩int(S))∪C for all B∈branch(S),C∈branch(TB). Clearly g((B,C)) is a chain.
To show maximality of g((B,C)), if a∈T such that a≤b or b≤a for all b∈g((B,C)), then either a∈TB which implies a∈C, or a∈int(S) which implies a∈B.
Clearly g is injective. To prove subjectivity, for any D∈branch(T), there is some B∈branch(S) such that D∩TB=∅. It follows that D∩TB is a maximal chain in TB, so g((B,D∩TB))=D. In conclusion g is bijective.
To prove g is an isomorphism. Let R∈L be a k-ary relation. Let (B1,C1),…,(Bk,Ck)∈∏SMs[∏TBMt]B∈branch(S).
Then exactly one of the following cases holds:
•
⋀1≤i<j≤kBi=Bj, in which case C1,…,Ck∈branch(TB1), so TB1∋m:=C1∧⋯∧Ck=g(B1,C1)∧⋯∧g(Bk,Ck) and
[TABLE]
•
⋁1≤i<j≤kBi=Bj, in which case S∋m:=B1∧⋯∧Bk=g(B1,C1)∧⋯∧g(Bk,Ck) and
[TABLE]
Finally, to prove g({B}×∏TBMt)={D∈∏TMt}D∩TB=∅ for all B∈branch(S), If B∈branch(S),C∈∏TBMt, let D:=g((B,C))=(B∩int(S))∪C and D∩TB′=[(B∩int(S))∪C]∩TB′=C∩TB′, and the latter is non-empty if and only if C∈∏TB′Mt which happens exactly when B=B′. In fact, if C∩TB=∅, then C⊆TB, therefore D∩TB=C∩TB=C. So if D∩TB=∅, then f(D)=(B,D∩TB).
∎
Lemma 4.16**.**
Let T be a successor meet tree, ⟨T,Mt⟩, ⟨T,Nt⟩ family trees, and t0∈T, such that Mt0≺Nt0 and Mt=Nt for all to=t∈T. Then ∏TMt≺∏TNt.
Proof.
Let A⊂T be any maximal anti-chain such that t0∈A. By Remark 4.12 and Lemma 4.15, we have
[TABLE]
Let P:=∏T≤AMt, P′:=∏T≤ANt, SB:=∏T≥sup(B)Mt, and SB′:=∏T≥sup(B)Nt for all B∈branch(T≤A).
So ∏TMt≅P[SB]B∈P and ∏TNt≅P′[SB′]B∈P.
Now there is some B0∈branch(T≤A) such that {t0}=B0∩A and t0=sup(B0). Since Mt=Nt for all t=t0, it follows that
[TABLE]
and also
[TABLE]
for all B0=B∈branch(B). If ∏T≥t0Mt≺∏T≥t0Nt,
then SB0≺SB0′ and, by Proposition 2.10,
[TABLE]
So it suffices to show ∏T≥t0Mt≺∏T≥t0Nt. If T≥t0 is a tree of finite height, then the claim follows from Proposition 2.10.
Otherwise, let S:=T≤t0 and let A′:=succ(t0). Notice that A′ is a maximal anti-chain in S and t0∈S≤A′. Moreover, S≤A′ is a tree of finite height. Again, by Remark 4.12, we have
[TABLE]
In this case, t0∈S≤A′, therefore ∏S≥sup(B)Mt=∏S≥sup(B)Nt for all B∈branch(S≤A′). So it suffices to show that ∏S≤A′Mt≺∏S≤A′Nt, but this follows from Remark 3.2.
∎
Corollary 4.17**.**
Let T be a successor meet tree, and let ⟨T,Mt⟩, ⟨T,Nt⟩ be family trees.
(1)
If Mt≺Nt for all t∈T, then ∏TMt≺∏TNt.
2. (2)
If Mt≡Nt for all t∈T, then ∏TMt≡∏TNt.
Proof.
Item 1 is by Lemma 4.16 and induction. For Item 2, for each t∈T, let Mt be a sufficiently saturated model of Th(Mt). Then there are elementary embeddings Mt,Nt↪Mt. By Item 1 of this corollary, we can find elementary embeddings ∏TMt,∏TNt↪∏TMt.
∎
5. Countable tree products
In this section we restrict ourself to the case where the structures in the product, as well as the trees themselves, are all countable.
Furthermore, as in the case of trees of finite height, we assume for simplicity all successor meet trees are leveled, i.e., any two branches have the same order type.
Remark 5.1**.**
If ⟨T,<⟩ is a leveled countable tree of infinite height such that ∣succ(a)∣=ℵ0 for all a∈int(T), then exactly one of the following holds.
(1)
Any branch is of order type ω, which by Lemma 4.8, is isomorphic to Example 4.6.
2. (2)
Any branch is of order type ω∗, which by Lemma 4.8, is isomorphic to Example 4.7.
3. (3)
Any branch is of order type Z, which by Lemma 4.8 and Lemma 4.15 is isomorphic to a finite product of the first two cases.
In each of the cases above, there is a canonical definition of height for any element of T, as follows:
Definition 5.2**.**
If T=ω<ω or T=S from Example 4.7 and t∈T, then height(t) is defined to be the maximum of the domain of t; i.e., if t=(7,3,2,8)∈ω<ω then height(t)=3, if t=(0,…,0,3,2,17,−,−,−)∈S then height(t)=−4. For t=()∈ω<ω we set height(t):=−1.
If T=S∗⟨TB⟩B∈branch(S) where S is as in Example 4.7 and TB=ω<ω, then height(t) is well defined and furthermore, height(t1)<height(t2) for all t1<t2∈T and succ(height(t1))=height(t2)⟺t2∈succ(t1).
We can now expand any countable product by infinitely many equivalence relations, in the same fashion as in Definition 1.1:
Definition 5.3**.**
Let ⟨T,Mt⟩ be a family tree. Let (∏TMt)s be an expansion of ∏TMt by binary relation symbols {sn}n∈Z interpreted as:
[TABLE]
Remark 5.4**.**
Let ⟨T,Mt⟩ be a family tree. Let Mts be Mt expanded by binary relation symbols {sn}n∈Z interpreted as:
[TABLE]
Then (∏TMt)s≅∏TMts.
For this reason, we identify the two constructions and denote the two by ∏TMts.**
Remark 5.5**.**
If Mt is transitive for all t∈T and aˉ is a tuple in ∏TMts then
[TABLE]
5.1. Dense substructures in countable tree products
If ⟨T,Mt⟩ is a family tree where every branch in T is of order type ω∗ (e.g., Example 4.7), then ∏TMt is countable. On the other hand, if every branch in T is of order type ω, then ∣∏TMt∣=2ℵ0. In order to keep the size of a product of any countable tree of countable structures to be countable, we introduce the notion of a dense substructure. A dense substructure may be countable, and as an induced substructure will be elementarily equivalent to the product, as will follow from Corollary 5.13. The main result of this subsection, Theorem 5.11, states that under certain homogeneity assumptions on the structures of a countable family tree ⟨T,Mt⟩ dense substructures of the product are homogeneous, and unique up to isomorphism; the precise assumption on ⟨T,Mt⟩ is that it is pure, as defined in Definition 5.9.
Definition 5.6**.**
Let ⟨T,Mt⟩ be a family tree. A substructure D⊆∏t∈TMts is dense
if for all t∈T, there is some d∈D such that d∋t.
Clearly whenever T is countable there is a countable dense substructure.
Remark 5.7**.**
Let ⟨T,Mt⟩ be a family tree and let A⊂T be a maximal anti-chain.
A substructure N⊆∏t∈TMts is dense
if and only if
for all a∈A and for all t≥a, there is some d∈N such that d∋t.
Remark 5.8**.**
If every branch in T is of order type ω∗, and ⟨T,Mt⟩ is a family tree with N⊆∏t∈TMts dense then N=∏t∈TMts.
Before continuing, we define a special kind of family tree that will be central throughout this subsection:
Definition 5.9**.**
A family tree ⟨T,Mt⟩ is pure if Mt is transitive and homogeneous for all t∈T and height(t)=height(u)⟹Mt≅Mu for all t,u∈T. It is ω-pure if, in addition, branches in T are of order type ω.
Lemma 5.10**.**
Let ⟨T,Mt⟩ be a pure family tree. If N⊆∏TMts is a countable dense substructure, then:
(1)
For any countable A⊆∏TMts, there is A′⊆N such that A≅A′.
2. (2)
N* is transitive and homogeneous.*
Proof.
Let aˉ,b∈∏TMts and cˉ∈N where aˉ,cˉ are finite tuples and tpqf(aˉ)=tpqf(cˉ). To prove both Item 1 and 2, it suffices to find some d∈N such that tpqf(aˉ,b)=tpqf(cˉ,d). If aˉ=cˉ=∅ then by Remark 5.5, for any d∈N, the mapping b↦d is a partial isomorphism. Otherwise, let f:aˉ→cˉ be a partial isomorphism. Let
t0:=max{a∧b}a∈aˉ. Notice that unless b∈aˉ, in which case the proof is trivial, t0 exists, as a maximum of finite elements in the chain b.
Let height(t0)=m. Let A0:={a∈aˉ}a∋t0. Notice that A0 is the sm-equivalence class of b in aˉ and A0=∅. Then f(A0) is also an sm equivalence class in cˉ, so there is some t1∈T with height(t1)=m such that f(A0)={c∈cˉ}c∋t1. Since Mt0≅Mt1 and Mt1 is homogeneous, it follows that there is some s∈Mt1 such that
[TABLE]
By density of N, there is some d∈N such that d∋s and and therefore, by Remark 5.5, tpqf(aˉ,b)=tpqf(cˉ,d).
∎
Theorem 5.11**.**
Let ⟨T,Mt⟩ be a pure family tree.
(1)
Up to isomorphism, there is a unique countable dense substructure D⊆∏TMts.
2. (2)
Such a D is transitive and homogeneous.
Proof.
Let N1,N2⊆∏TMts be two countable dense substructures. By Lemma 5.10, they are both transitive homogeneous, so to prove both 1 and 2 it is left to show that N1≅N2. For that, by Lemma 5.10, every substructure A⊆N1 is embeddable in N2 and vice-versa. Using this fact and homogeneity, a standard back-and-forth argument yields an isomorphism between N1 and N2.
∎
Corollary 5.12**.**
Let ⟨T,Mt⟩ and ⟨U,Nu⟩ be isomorphic pure trees.
If D1,D2 are countable dense substructures in ∏TMts,∏UNus respectively, then D1≅D2.
Proof.
By Remark 4.10, ∏TMts≅∏UNus, so D2 is isomorphic (via the restriction of an isomorphism) to a dense substructure of ∏TMts, which in turn, by Theorem 5.11, is isomorphic to D1.
∎
Corollary 5.13**.**
Let ⟨T,Mt⟩ be a pure family tree.
If D1,D2⊆∏TMts are dense then D1≡ω1,ωD2.
Proof.
Let D01⊆D1, D02⊆D2 be countable dense substructures. By downwards Löwenheim-Skolem for Lω1,ω, there are countable A1,A2 such that D01⊆A1⪯ω1,ωD1 and D02⊆A2⪯ω1,ωD2. Since D01 and D02 are dense, so are A1 and A2. Therefore, by Theorem 5.11, A1≅A2. In conclusion,
D1⪰ω1,ωA1≅A2⪯ω1,ωD2.
∎
Notation 5.14**.**
For L-structures M and N, we denote M∼eN if M can be elementarily embedded in N and vice-versa.
Lemma 5.15**.**
Let ⟨T,Mt⟩ and ⟨T,Nt⟩ be family trees such that ⟨T,Mt⟩ is pure.
If Nt⪰Mt for all t∈T, then for any countable dense substructure D1⊆∏TMts there is some countable dense elementary substructure D2⪯∏TNts such that D1 embeds elementarily into D2.
Proof.
By downwards Löwenheim-Skolem, there is some countable dense elementary substructure D1′≺∏TMts. By Theorem 5.11, D1≅D1′, so we may assume D1≺∏TMts. Now by Corollary 4.17, there is an elementary embedding e:∏TMts↪∏TNts. Again, by Löwenheim-Skolem, there is some countable dense elementary substructure e(D1)⊆D2≺∏TNts. So if ι is the inclusion map we have the following commutative diagram:
Let ⟨T,Mt⟩ and ⟨T,Nt⟩ be family trees such that ⟨T,Mt⟩ is pure.
If Nt⪯Mt for all t∈T, then for any countable dense substructure D1⊂∏TMts and any countable dense elementary substructure D2≺∏TNts there is an elementary embedding of D2 into D1.
Proof.
By Corollary 4.17, there is an elementary embedding e:∏TNts↪∏TMts. By Löwenheim-Skolem, there is some countable dense elementary substructure e(D2)⊆D1′≺∏TMts. So if ι is the inclusion map we have the following commutative diagram:
So e:D2↪D1′ is elementary. Now by Theorem 5.11, D1′≅D1.
∎
Corollary 5.17**.**
Let ⟨T,Mt⟩ and ⟨T,Nt⟩ be family trees such that ⟨T,Mt⟩ is pure.
If Nt∼eMt for all t∈T, then for any countable dense substructure D1⊂∏TMts there is some countable dense elementary substructure D2≺∏TNts such that D1∼eD2.
Proof.
By Lemma 5.15, we can find some countable dense elementary substructure D2≺∏TNts such that D1 elementarily embeds into D2. By Lemma 5.16, D2 elementarily embeds into D1.
∎
Lemma 5.18**.**
Let ⟨T,Mt⟩ be a countable family tree and A⊂T a maximal anti-chain. Then there is an isomorphism
[TABLE]
such that
(1)
f({D∈N}D∩T≥sup(B)=∅)=f(N)∩({B}×∏T≥sup(B)Mts)* for all N⊂∏TMts,B∈branch(T≤A).*
2. (2)
If D⊆∏TMts is dense then for every B∈branch(T≤A), there is a dense DB⊆∏TBMt such that
f(\mathcal{D})=\prod_{T_{\leq A}}\mathcal{M}_{t}\Big{[}D_{B}\Big{]}_{B\in\operatorname{branch}(T_{\leq A})}
3. (3)
Conversly, if DB⊆∏TBMt is dense for every B∈branch(T≤A), then
So for every D⊂∏TMts and every B∈branch(T≤A) there is some DB⊇∏T≥sup(B)Mts such that
[TABLE]
So f(\mathcal{D})=\prod_{T_{\leq A}}\mathcal{M}_{t}\Big{[}\mathcal{D}_{B}\Big{]}_{B\in\operatorname{branch}(T_{\leq A}).}
To prove both 2 and 3, by Remark 5.7, D is dense if and only if for any a∈A and t≥a0, there is some d∈D such that d∋t.
2. (2)
Assume DB is dense for all B∈branch(T≤A). For any a0∈A and t≥a0, there is some B0∈branch(T≤A) such that a0=sup(B0). In particular, t∈TB0 and by density of DB0, there is some C0∈DB0 such that C0∋t. Now let d:=f−1(B0,C0). Then C0=d∩TB0. in particular, t0∈C0⊆d.
3. (3)
If D is dense, given t∈TB, by density of D, there is some d∈D such that d∋t. Now let (B,C):=f(d). Then C∈DB and C=d∩TB∋t.
∎
Corollary 5.19**.**
Let ⟨T,Mt⟩ be a countable family tree and let D⊆∏TMts be dense. Then for any t0∈T, the substructure induced on D∋t0:={d∈D}d∋t0 is isomorphic to some dense substructure D0 of ∏T≥t0Mt.
Proof.
Let A be a maximal anti-chain such that t0∈A, then there is some B0∈branch(T≤A) such that t0=sup(B0). Notice that D∋t0={d∈D}d∩T≥sup(B0)=∅.
Let
[TABLE]
be an isomorphism provided by Lemma 5.18. Then for every B∈branch(T≤A), there is a dense DB⊆∏TBMt such that
f(\mathcal{D})=\prod_{T_{\leq A}}\mathcal{M}_{t}\Big{[}D_{B}\Big{]}_{B\in\operatorname{branch}(T_{\leq A})}. Thus
[TABLE]
∎
6. (Elementary) indivisibility of infinite tree products
Recall a first-order relational structure is elementarily indivisible if for every colouring of its universe in two colours, there is a monochromatic elementary substructure isomorphic to it.
The aim of this section is to prove the following theorem, and to utilize it to give an example of a rigid elementarily indivisible structure, giving a negative answer to 1.2. This, together with [Mei16], completes answering all questions from [HKO11].
Theorem 6.1**.**
Let ⟨T,Mt⟩ be an ω-pure family tree, where Mt is indivisible for all t∈T.
If D⊂∏TMts be a countable dense substructure, then D is elementarily indivisible.
Proof.
By Theorem 5.11, Item 2, D is homogeneous, so by Lemma 2.7 indivisibility and elementary indivisibility coincide. To prove indivisibility, let c:D→{red,blue}. By Corollary 5.12, it suffices to find a subtree S⊂T and a tree isomorphism θ:S→T, such that θ↾Ms:Ms→Mθ(s) is an isomorphism of L structures, and a countable dense monochromatic substructure D2⊂∏SMt.
For every t∈T, let D∋t:={a∈D}a∋t. So c induces a sub-colouring of D∋t. We colour T as follows:
[TABLE]
If C(root(T))=blue then we are done. Otherwise, we continue constructing a C-red S and θ:S→T by induction on height(t):
(1)
S0=root(T);θ0=(root(T),root(T)).
2. (2)
Assume C(t)=red for all t∈Sn and let s∈Sn. by indivisibility of Ms, either B(s):={t∈succ(s)}C(t)=blue or R(s):={t∈succ(s)}C(t)=red contains an isomorphic copy of Ms.
•
If B(s) contains an isomorphic copy of Ms, denote it by Ms′, then D∋u contains an isomorphic monochromatic-blue copy D∋u′ of itself for every u∈Ms′.
By Corollary 5.19, for every u∈T, there is some dense substructure Du⊆∏T≥uMt such that Du≅D∋u. Let S:=T≥s. By Lemma 5.18 and Theorem 5.11,
[TABLE]
sup(B)∈succ(s) for every B∈branch(S≤succ(s)).
By Remark 3.2, ∏S≤succ(s)Mts≅Mss, so together with Equation 1,
[TABLE]
On the other hand, notice that the induced substructure on ⋃u∈Ms′D∋u′⊆D∋s is isomorphic to Ms′s[D∋u′]u∈Ms′, which in turn, by Equation 2 is isomorphic to Ds≅D∋s. So D∋s contains an isomorphic monochromatic-blue copy of itself, by contradiction to the induction hypothesis.
•
So R(s) contains an isomorphic copy of Ms, denoted by Ms′. Let θs:Ms′→Ms be such an isomorphism.
To conclude we define Sn+1:=⋃s∈SnMs′ and θn+1:=⋃s∈Sαθs
If S=⋃n<ωSn and θ=⋃n<ωθα, then by its construction θ:S→T is an isomorphism of trees such that θ↾Ms′:Ms′→Mθ(s) is an isomorphism of L structures. Since C(s)=red for all s∈S, by definition of C, there is a countable dense monochromatic-red D2⊂∏SMs.
∎
Theorem 6.2**.**
There is a countable rigid elementarily indivisible structure, in a finite language.
For the proof of Theorem 6.2, we will need the following:
There is a sequence {Ai}i∈ω of pairwise-non-isomorphic countable elementarily indivisible structures, in a finite language, such that Ai≺Aj for all i,j∈ω. Furthermore, A0 can be chosen to be homogeneous.
We first give an example in an infinite language and then present a structure in a finite language that is interdefinable with the first, i.e., a structure on the same underlying set with the same ∅-definable sets.
For the first example, in an infinite language: Let {Ai}i∈ω be a set of pairwise-non-isomorphic countable elementarily indivisible structures, in a finite language L, such that Ai∼eAj for all i,j∈ω such that A0 is homogeneous, as provided by 6.4.
Let T=ω<ω. Let ⟨σi⟩i∈ω be an enumeration of T. Let Ma0:=A0 for all a∈T and Nσi0:=Ai. For all a∈T, let Ma and Na be expansions of Ma0 and Na0, respectively, to a new binary relation R such that Ma and Na both interpret R as a full subgraph whenever height(a) is even and as an empty subgraph whenever height(a) is odd.
Let D′⊂∏TMts be countable and dense.
By Theorem 6.1, D′ is elementarily indivisible. By Corollary 4.17, ∏TsMt∼e∏TsNt.
By Corollary 5.17 there is a countable dense elementary substructure D≺∏TNts such that D′∼eD. Since D′ is elementarily indivisible, so is D, by 6.3. Now D is rigid since if there are distinct a,b∈D and σ∈Aut(D) such that σ(a)=b, since a=b, there is some i<ω such that ¬si(a,b) but σ sends the si-equivalence class of a to the si-equivalence class of b, but, by definition of D, no two si-equivalence classes are isomorphic.
For an example in a finite language, we notice that si is definable from R, for all 1≤i<ω:
s_{2n}(x,y)\leftrightarrow\Bigg{(}s_{2n-1}(x,y)\land\bigg{(}R(x,y)\lor\exists z\Big{(}s_{2n-1}(x,z)\land R(x,z)\land R(y,z)\Big{)}\bigg{)}\Bigg{)} for n≥1.
•
s_{2n+1}(x,y)\leftrightarrow\Bigg{(}s_{2n}(x,y)\land\bigg{(}\neg R(x,y)\lor\exists z\Big{(}s_{2n}(x,z)\land\neg R(x,z)\land\neg R(y,z)\Big{)}\bigg{)}\Bigg{)} for n≥1.
So D and D↾L∪{R} are inter-definable, the latter being in a finite language.
∎
6.1. Lω1,ω-elementary indivisibility and transitivity
In this subsection, we strengthen the notion of elementary indivisibility to Lω1,ω and show that not only does Theorem 6.2 fail in this context, but in fact, every Lω1,ω-elementarily indivisible structure is transitive.
Definition 6.5**.**
A relational structure is Lω1,ω-elementarily indivisible if for every colouring of its universe in two colours, there is a monochromatic Lω1,ω-elementary substructure isomorphic to it.
Lemma 6.6**.**
If M is a countable Lω1,ω-elementarily indivisible structure then a≡ω1,ωb for any two singletons a,b∈M.
Proof.
If not, then there is an Lω1,ω-formula in one free variable ϕ(x) such that M⊨∃xϕ(x) and M⊨∃x¬ϕ(x). Let c:M→{red,blue} be defined as
[TABLE]
Clearly, no c-monochromatic substructure is Lω1,ω-elementary.
∎
Theorem 6.7**.**
Every countable Lω1,ω-elementarily indivisible structure is transitive.
Proof.
Let M be an Lω1,ω-elementarily indivisible structure and let a,b∈M be singletons, then by Lemma 6.6, a≡ω1,ωb. By Scott’s Isomorphism Theorem ([Sco65], [Hod93, Corollary 3.5.4]), ⟨M,a⟩≅⟨M,b⟩ (where ⟨M,a⟩,⟨M,b⟩ are expansions of M by a constant symbol for a,b respectively). Finally, any isomorphism between ⟨M,a⟩ and ⟨M,b⟩ is an automorphism of M sending a to b.
∎
Bibliography10
The reference list from the paper itself. Each links out to its DOI / PubMed record.
1[Che 98] Gregory L. Cherlin. The classification of countable homogeneous directed graphs and countable homogeneous n 𝑛 n -tournaments. Mem. Amer. Math. Soc. , 131(621):xiv+161, 1998.
2[EZS 93] Mohamed M. El-Zahar and Norbert W. Sauer. On the divisibility of homogeneous directed graphs. Canad. J. Math. , 45(2):284–294, 1993.
3[EZS 94] Mohamed M. El-Zahar and Norbert W. Sauer. On the divisibility of homogeneous hypergraphs. Combinatorica , 14(2):159–165, 1994.
4[Fra 00] Roland Fraïssé. Theory of relations , volume 145 of Studies in Logic and the Foundations of Mathematics . North-Holland Publishing Co., Amsterdam, revised edition, 2000. With an appendix by Norbert Sauer.
5[HKO 11] Assaf Hasson, Menachem Kojman, and Alf Onshuus. On symmetric indivisibility of countable structures. In Model theoretic methods in finite combinatorics , volume 558 of Contemp. Math. , pages 417–452. Amer. Math. Soc., Providence, RI, 2011.
6[Hod 93] Wilfrid Hodges. Model theory , volume 42 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1993.
7[KR 86] Péter Komjáth and Vojtěch Rödl. Coloring of universal graphs. Graphs Combin. , 2(1):55–60, 1986.
8[Lac 87] A. H. Lachlan. Homogeneous structures. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) , pages 314–321. Amer. Math. Soc., Providence, RI, 1987.