Ranks for families of theories and their spectra

TL;DR

This paper introduces ranks and degrees for families of theories, extending classical notions like Morley rank, and explores their properties, bounds, and criteria for total transcendentality within model theory.

Contribution

It defines new rank and degree notions for families of theories, analyzes their preservation under E-closures, and establishes criteria for total transcendentality in countable languages.

Findings

Bounds for e-spectra with respect to ranks and degrees

Ranks and degrees are preserved under E-closures

Criteria for total transcendental families based on cardinality

Abstract

We define ranks and degrees for families of theories, similar to Morley rank and degree, as well as Cantor-Bendixson rank and degree, and the notion of totally transcendental family of theories. Bounds for $e$-spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under $E$-closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of $E$-closure and of the $e$-spectrum value, for a countable language, are proved.

Equations

$$e\text{mbox}-\mathrm{Sp}({\mathcal{T}}_0)=e\text{mbox}-\mathrm{Sp}({\mathcal{T}}_0^{\prime })=|{\mathcal{T}}_0\setminus {\mathcal{T}}_0^{\prime }|.$$

$$\mathrm{RS}({\mathrm{Cl}}_E(\mathcal{T}))\le \mathrm{RS}(\mathcal{T})$$

Full text

Key words: family of theories, rank, degree, $E$-closure, $e$-spectrum.

We continue to study families of theories [1, 2, 3, 4, 5, 6] and their approximations [7] introducing ranks and degrees for families of theories, similar to Morley rank and degree [8], as well as Cantor-Bendixson rank and degree, and the notion of totally transcendental family of theories. These ranks and degree plays a similar role for families of theories, with hierarchies for definable sets of theories, as Morley ones for a fixed theory although they have own specificities.

Bounds for $e$-spectra with respect to ranks and degrees are found. It is shown that the ranks and the degrees are preserved under $E$-closures and values for the ranks and the degrees are characterized. Criteria for totally transcendental families in terms of cardinality of $E$-closure and of the $e$-spectrum value, for a countable language, are proved.

1 Preliminaries

Throughout the paper we consider complete first-order theories $T$ in predicate languages $\Sigma(T)$ and use the following terminology in [1, 2, 3, 4, 5, 6].

Let $P=(P_{i})_{i\in I}$, be a family of nonempty unary predicates, $(\mathcal{A}_{i})_{i\in I}$ be a family of structures such that $P_{i}$ is the universe of $\mathcal{A}_{i}$, $i\in I$, and the symbols $P_{i}$ are disjoint with languages for the structures $\mathcal{A}_{j}$, $j\in I$. The structure $\mathcal{A}_{P}\rightleftharpoons\bigcup\limits_{i\in I}\mathcal{A}_{i}$ expanded by the predicates $P_{i}$ is the $P$-union of the structures $\mathcal{A}_{i}$, and the operator mapping $(\mathcal{A}_{i})_{i\in I}$ to $\mathcal{A}_{P}$ is the $P$-operator. The structure $\mathcal{A}_{P}$ is called the $P$-combination of the structures $\mathcal{A}_{i}$ and denoted by ${\rm Comb}_{P}(\mathcal{A}_{i})_{i\in I}$ if $\mathcal{A}_{i}=(\mathcal{A}_{P}\upharpoonright A_{i})\upharpoonright\Sigma(\mathcal{A}_{i})$, $i\in I$. Structures $\mathcal{A}^{\prime}$, which are elementary equivalent to ${\rm Comb}_{P}(\mathcal{A}_{i})_{i\in I}$, will be also considered as $P$-combinations.

Clearly, all structures $\mathcal{A}^{\prime}\equiv{\rm Comb}_{P}(\mathcal{A}_{i})_{i\in I}$ are represented as unions of their restrictions $\mathcal{A}^{\prime}_{i}=(\mathcal{A}^{\prime}\upharpoonright P_{i})\upharpoonright\Sigma(\mathcal{A}_{i})$ if and only if the set $p_{\infty}(x)=\{\neg P_{i}(x)\mid i\in I\}$ is inconsistent. If $\mathcal{A}^{\prime}\neq{\rm Comb}_{P}(\mathcal{A}^{\prime}_{i})_{i\in I}$, we write $\mathcal{A}^{\prime}={\rm Comb}_{P}(\mathcal{A}^{\prime}_{i})_{i\in I\cup\{\infty\}}$, where $\mathcal{A}^{\prime}_{\infty}=\mathcal{A}^{\prime}\upharpoonright\bigcap\limits_{i\in I}\overline{P_{i}}$, maybe applying Morleyzation. Moreover, we write ${\rm Comb}_{P}(\mathcal{A}_{i})_{i\in I\cup\{\infty\}}$ for ${\rm Comb}_{P}(\mathcal{A}_{i})_{i\in I}$ with the empty structure $\mathcal{A}_{\infty}$.

Note that if all predicates $P_{i}$ are disjoint, a structure $\mathcal{A}_{P}$ is a $P$-combination and a disjoint union of structures $\mathcal{A}_{i}$. In this case the $P$-combination $\mathcal{A}_{P}$ is called disjoint. Clearly, for any disjoint $P$-combination $\mathcal{A}_{P}$, ${\rm Th}(\mathcal{A}_{P})={\rm Th}(\mathcal{A}^{\prime}_{P})$, where $\mathcal{A}^{\prime}_{P}$ is obtained from $\mathcal{A}_{P}$ replacing $\mathcal{A}_{i}$ by pairwise disjoint $\mathcal{A}^{\prime}_{i}\equiv\mathcal{A}_{i}$, $i\in I$. Thus, in this case, similar to structures the $P$-operator works for the theories $T_{i}={\rm Th}(\mathcal{A}_{i})$ producing the theory $T_{P}={\rm Th}(\mathcal{A}_{P})$, being $P$-combination of $T_{i}$, which is denoted by ${\rm Comb}_{P}(T_{i})_{i\in I}$.

Notice that $P$-combinations are represented by generalized products of structures [9].

For an equivalence relation $E$ replacing disjoint predicates $P_{i}$ by $E$-classes we get the structure $\mathcal{A}_{E}$ being the $E$-union of the structures $\mathcal{A}_{i}$. In this case the operator mapping $(\mathcal{A}_{i})_{i\in I}$ to $\mathcal{A}_{E}$ is the $E$-operator. The structure $\mathcal{A}_{E}$ is also called the $E$-combination of the structures $\mathcal{A}_{i}$ and denoted by ${\rm Comb}_{E}(\mathcal{A}_{i})_{i\in I}$; here $\mathcal{A}_{i}=(\mathcal{A}_{E}\upharpoonright A_{i})\upharpoonright\Sigma(\mathcal{A}_{i})$, $i\in I$. Similar above, structures $\mathcal{A}^{\prime}$, which are elementary equivalent to $\mathcal{A}_{E}$, are denoted by ${\rm Comb}_{E}(\mathcal{A}^{\prime}_{j})_{j\in J}$, where $\mathcal{A}^{\prime}_{j}$ are restrictions of $\mathcal{A}^{\prime}$ to its $E$-classes. The $E$-operator works for the theories $T_{i}={\rm Th}(\mathcal{A}_{i})$ producing the theory $T_{E}={\rm Th}(\mathcal{A}_{E})$, being $E$-combination of $T_{i}$, which is denoted by ${\rm Comb}_{E}(T_{i})_{i\in I}$ or by ${\rm Comb}_{E}(\mathcal{T})$, where $\mathcal{T}=\{T_{i}\mid i\in I\}$.

Clearly, $\mathcal{A}^{\prime}\equiv\mathcal{A}_{P}$ realizing $p_{\infty}(x)$ is not elementary embeddable into $\mathcal{A}_{P}$ and can not be represented as a disjoint $P$-combination of $\mathcal{A}^{\prime}_{i}\equiv\mathcal{A}_{i}$, $i\in I$. At the same time, there are $E$-combinations such that all $\mathcal{A}^{\prime}\equiv\mathcal{A}_{E}$ can be represented as $E$-combinations of some $\mathcal{A}^{\prime}_{j}\equiv\mathcal{A}_{i}$. We call this representability of $\mathcal{A}^{\prime}$ to be the $E$-representability.

If there is $\mathcal{A}^{\prime}\equiv\mathcal{A}_{E}$ which is not $E$-representable, we have the $E^{\prime}$-representability replacing $E$ by $E^{\prime}$ such that $E^{\prime}$ is obtained from $E$ adding equivalence classes with models for all theories $T$, where $T$ is a theory of a restriction $\mathcal{B}$ of a structure $\mathcal{A}^{\prime}\equiv\mathcal{A}_{E}$ to some $E$-class and $\mathcal{B}$ is not elementary equivalent to the structures $\mathcal{A}_{i}$. The resulting structure $\mathcal{A}_{E^{\prime}}$ (with the $E^{\prime}$-representability) is a $e$-completion, or a $e$-saturation, of $\mathcal{A}_{E}$. The structure $\mathcal{A}_{E^{\prime}}$ itself is called $e$-complete, or $e$-saturated, or $e$-universal, or $e$-largest.

For a structure $\mathcal{A}_{E}$ the number of new structures with respect to the structures $\mathcal{A}_{i}$, i. e., of the structures $\mathcal{B}$ which are pairwise elementary non-equivalent and elementary non-equivalent to the structures $\mathcal{A}_{i}$, is called the $e$-spectrum of $\mathcal{A}_{E}$ and denoted by $e$-${\rm Sp}(\mathcal{A}_{E})$. The value ${\rm sup}\{e$-${\rm Sp}(\mathcal{A}^{\prime}))\mid\mathcal{A}^{\prime}\equiv\mathcal{A}_{E}\}$ is called the $e$-spectrum of the theory ${\rm Th}(\mathcal{A}_{E})$ and denoted by $e$-${\rm Sp}({\rm Th}(\mathcal{A}_{E}))$. If structures $\mathcal{A}_{i}$ represent theories $T_{i}$ of a family $\mathcal{T}$, consisting of $T_{i}$, $i\in I$, then the $e$-spectrum $e$-${\rm Sp}(\mathcal{A}_{E})$ is denoted by $e$-${\rm Sp}(\mathcal{T})$.

If $\mathcal{A}_{E}$ does not have $E$-classes $\mathcal{A}_{i}$, which can be removed, with all $E$-classes $\mathcal{A}_{j}\equiv\mathcal{A}_{i}$, preserving the theory ${\rm Th}(\mathcal{A}_{E})$, then $\mathcal{A}_{E}$ is called $e$-prime, or $e$-minimal.

For a structure $\mathcal{A}^{\prime}\equiv\mathcal{A}_{E}$ we denote by ${\rm TH}(\mathcal{A}^{\prime})$ the set of all theories ${\rm Th}(\mathcal{A}_{i})$ of $E$-classes $\mathcal{A}_{i}$ in $\mathcal{A}^{\prime}$.

By the definition, an $e$-minimal structure $\mathcal{A}^{\prime}$ consists of $E$-classes with a minimal set ${\rm TH}(\mathcal{A}^{\prime})$. If ${\rm TH}(\mathcal{A}^{\prime})$ is the least for models of ${\rm Th}(\mathcal{A}^{\prime})$ then $\mathcal{A}^{\prime}$ is called $e$-least.

Definition [2]. Let $\overline{\mathcal{T}}_{\Sigma}$ be the set of all complete elementary theories of a relational language $\Sigma$. For a set $\mathcal{T}\subset\overline{\mathcal{T}}_{\Sigma}$ we denote by ${\rm Cl}_{E}(\mathcal{T})$ the set of all theories ${\rm Th}(\mathcal{A})$, where $\mathcal{A}$ is a structure of some $E$-class in $\mathcal{A}^{\prime}\equiv\mathcal{A}_{E}$, $\mathcal{A}_{E}={\rm Comb}_{E}(\mathcal{A}_{i})_{i\in I}$, ${\rm Th}(\mathcal{A}_{i})\in\mathcal{T}$. As usual, if $\mathcal{T}={\rm Cl}_{E}(\mathcal{T})$ then $\mathcal{T}$ is said to be $E$-closed.

The operator ${\rm Cl}_{E}$ of $E$-closure can be naturally extended to the classes $\mathcal{T}\subset\overline{\mathcal{T}}$, where $\overline{\mathcal{T}}$ is the union of all $\overline{\mathcal{T}}_{\Sigma}$ as follows: ${\rm Cl}_{E}(\mathcal{T})$ is the union of all ${\rm Cl}_{E}(\mathcal{T}_{0})$ for subsets $\mathcal{T}_{0}\subseteq\mathcal{T}$, where new language symbols with respect to the theories in $\mathcal{T}_{0}$ are empty.

For a set $\mathcal{T}\subset\overline{\mathcal{T}}$ of theories in a language $\Sigma$ and for a sentence $\varphi$ with $\Sigma(\varphi)\subseteq\Sigma$ we denote by $\mathcal{T}_{\varphi}$ the set $\{T\in\mathcal{T}\mid\varphi\in T\}$. Any set $\mathcal{T}_{\varphi}$ is called the $\varphi$-neighbourhood, or simply a neighbourhood, for $\mathcal{T}$, or the ($\varphi$-)definable subset of $\mathcal{T}$.

Proposition 1.1 [2]. If $\mathcal{T}\subset\overline{\mathcal{T}}$ is an infinite set and $T\in\overline{\mathcal{T}}\setminus\mathcal{T}$ then $T\in{\rm Cl}_{E}(\mathcal{T})$ (i.e., $T$ is an accumulation point for $\mathcal{T}$ with respect to $E$-closure ${\rm Cl}_{E}$) if and only if for any formula $\varphi\in T$ the set $\mathcal{T}_{\varphi}$ is infinite.

If $T$ is an accumulation point for $\mathcal{T}$ then we also say that $T$ is an accumulation point for ${\rm Cl}_{E}(\mathcal{T})$.

Theorem 1.2 [2]. For any sets $\mathcal{T}_{0},\mathcal{T}_{1}\subset\overline{\mathcal{T}}$, ${\rm Cl}_{E}(\mathcal{T}_{0}\cup\mathcal{T}_{1})={\rm Cl}_{E}(\mathcal{T}_{0})\cup{\rm Cl}_{E}(\mathcal{T}_{1})$.

Definition [2]. Let $\mathcal{T}_{0}$ be a closed set in a topological space $(\mathcal{T},\mathcal{O}_{E}(\mathcal{T}))$, where $\mathcal{O}_{E}(\mathcal{T})=\{\mathcal{T}\setminus{\rm Cl}_{E}(\mathcal{T}^{\prime})\mid\mathcal{T}^{\prime}\subseteq\mathcal{T}\}$. A subset $\mathcal{T}^{\prime}_{0}\subseteq\mathcal{T}_{0}$ is said to be generating if $\mathcal{T}_{0}={\rm Cl}_{E}(\mathcal{T}^{\prime}_{0})$. The generating set $\mathcal{T}^{\prime}_{0}$ (for $\mathcal{T}_{0}$) is minimal if $\mathcal{T}^{\prime}_{0}$ does not contain proper generating subsets. A minimal generating set $\mathcal{T}^{\prime}_{0}$ is least if $\mathcal{T}^{\prime}_{0}$ is contained in each generating set for $\mathcal{T}_{0}$.

Theorem 1.3 [2]. If $\mathcal{T}^{\prime}_{0}$ is a generating set for a $E$-closed set $\mathcal{T}_{0}$ then the following conditions are equivalent:

$(1)$* $\mathcal{T}^{\prime}_{0}$ is the least generating set for $\mathcal{T}_{0}$;*

$(2)$* $\mathcal{T}^{\prime}_{0}$ is a minimal generating set for $\mathcal{T}_{0}$;*

$(3)$* any theory in $\mathcal{T}^{\prime}_{0}$ is isolated by some set $(\mathcal{T}^{\prime}_{0})_{\varphi}$, i.e., for any $T\in\mathcal{T}^{\prime}_{0}$ there is $\varphi\in T$ such that $(\mathcal{T}^{\prime}_{0})_{\varphi}=\{T\}$;*

$(4)$* any theory in $\mathcal{T}^{\prime}_{0}$ is isolated by some set $(\mathcal{T}_{0})_{\varphi}$, i.e., for any $T\in\mathcal{T}^{\prime}_{0}$ there is $\varphi\in T$ such that $(\mathcal{T}_{0})_{\varphi}=\{T\}$.*

Notice that having the least generating set $\mathcal{T}^{\prime}_{0}$ for a $E$-closed set $\mathcal{T}_{0}$,

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Definition [7]. Let $\mathcal{T}$ be a class of theories and $T$ be a theory, $T\notin\mathcal{T}$. The theory $T$ is called $\mathcal{T}$-approximated, or approximated by $\mathcal{T}$, or $\mathcal{T}$-approximable, or a pseudo-$\mathcal{T}$-theory, if for any formula $\varphi\in T$ there is $T^{\prime}\in\mathcal{T}$ such that $\varphi\in T^{\prime}$.

If $T$ is $\mathcal{T}$-approximated then $\mathcal{T}$ is called an approximating family for $T$, theories $T^{\prime}\in\mathcal{T}$ are approximations for $T$, and $T$ is an accumulation point for $\mathcal{T}$.

An approximating family $\mathcal{T}$ is called single-valued, or $e$-categorical, if $e$-${\rm Sp}(\mathcal{T})=1$.

An approximating family $\mathcal{T}$ is called $e$-minimal if for any sentence $\varphi\in\Sigma(T)$, $\mathcal{T}_{\varphi}$ is finite or $\mathcal{T}_{\neg\varphi}$ is finite.

As in [7] we permit extensions of $e$-minimal / $e$-categorical families by their accumulation points and these extensions will be also called $e$-minimal / $e$-categorical.

Theorem 1.4 [7]. A family $\mathcal{T}$ is $e$-minimal if and only if it is $e$-categorical.

Proposition 1.5 [7]. Any $E$-closed family $\mathcal{T}$ with finite $e$-${\rm Sp}(\mathcal{T})>0$ is represented as a disjoint union of $e$-categorical families $\mathcal{T}_{1},\ldots,\mathcal{T}_{n}$.

Proof. Let $e$-${\rm Sp}(\mathcal{T})=n$ and $T_{1},\ldots,T_{n}$ be accumulation points for $\mathcal{T}$ witnessing that equality. Now we consider pairwise inconsistent formulas $\varphi_{i}\in T_{i}$ separating $T_{i}$ from $T_{j}$, $j\neq i$, i.e., with $\neg\varphi_{i}\in T_{j}$. By Proposition 1.1 each family $\mathcal{T}_{i}=\mathcal{T}_{\varphi_{i}}$ is infinite, with unique accumulation point $T_{i}$, and thus $\mathcal{T}_{i}$ is $e$-categorical. Besides, the families $\mathcal{T}_{i}$ are disjoint by the choice of $\varphi_{i}$, and $\mathcal{T}^{\prime}=\mathcal{T}\setminus\left(\bigcup\limits_{i=1}^{n}\mathcal{T}_{i}\right)$ does not have accumulation points. Therefore $\mathcal{T}^{\prime}\cup\mathcal{T}_{1}$ is $e$-categorical, too. Thus, $\mathcal{T}^{\prime}\cup\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{n}$ is the required partition of $\mathcal{T}$ on $e$-categorical families. $\Box$

Theorem 1.6 [7]. A family $\mathcal{T}$ of theories contains an approximating subfamily if and only if $\mathcal{T}$ is infinite.

Proof. Since any approximating family is infinite then, having an approximating subfamily, $\mathcal{T}$ is infinite.

Conversely, let $\mathcal{T}$ be infinite. Firstly, we assume that the language $\Sigma=\Sigma(\mathcal{T})$ of $\mathcal{T}$ is at most countable. We enumerate all $\Sigma$-sentences: $\varphi_{n}$, $n\in\omega$, and construct an accumulation point for $\mathcal{T}$ by induction. Since $\mathcal{T}_{\varphi_{0}}$ or $\mathcal{T}_{\neg\varphi_{0}}$ is infinite we can choose $\psi_{0}=\varphi^{\delta}_{0}$ with infinite $\mathcal{T}_{\varphi^{\delta}_{0}}$, $\delta\in\{0,1\}$. If $\psi_{n}$ is already defined, with infinite $\mathcal{T}_{\psi_{n}}$, then we choose $\psi_{n+1}=\psi_{n}\wedge\varphi^{\delta}_{n+1}$, with $\delta\in\{0,1\}$, such that $\mathcal{T}_{\psi_{n+1}}$ is infinite. Finally, the set $\{\psi_{n}\mid n\in\omega\}$ forces a complete theory $T$ being an accumulation point both for $\mathcal{T}$ and for each $\mathcal{T}_{\psi_{n}}$. Thus, $\mathcal{T}\setminus\{T\}$ is a required approximating family.

If $\Sigma$ is uncountable we find an accumulation point $T_{0}$ for infinite $\mathcal{T}\upharpoonright\Sigma_{0}$, where $\Sigma_{0}$ is a countable sublanguage of $\Sigma$. Now we extend $T_{0}$ till a complete $\Sigma$-theory $T$ adding $\Sigma$-sentences $\chi$ such that $\mathcal{T}_{\chi}$ are infinite. Again $\mathcal{T}\setminus\{T\}$ is a required approximating family. $\Box$

2 Ranks and $e$-spectra

Starting with $e$-categorical, i.e., $e$-minimal families of theories we define the rank ${\rm RS}(\cdot)$ for the families of theories, similar to Morley rank [8], and a hierarchy with respect to these ranks in the following way.

For the empty family $\mathcal{T}$ we put the rank ${\rm RS}(\mathcal{T})=-1$, for finite nonempty families $\mathcal{T}$ we put ${\rm RS}(\mathcal{T})=0$, and for infinite families $\mathcal{T}$ — ${\rm RS}(\mathcal{T})\geq 1$.

For a family $\mathcal{T}$ and an ordinal $\alpha=\beta+1$ we put ${\rm RS}(\mathcal{T})\geq\alpha$ if there are pairwise inconsistent $\Sigma(\mathcal{T})$-sentences $\varphi_{n}$, $n\in\omega$, such that ${\rm RS}(\mathcal{T}_{\varphi_{n}})\geq\beta$, $n\in\omega$.

If $\alpha$ is a limit ordinal then ${\rm RS}(\mathcal{T})\geq\alpha$ if ${\rm RS}(\mathcal{T})\geq\beta$ for any $\beta<\alpha$.

We set ${\rm RS}(\mathcal{T})=\alpha$ if ${\rm RS}(\mathcal{T})\geq\alpha$ and ${\rm RS}(\mathcal{T})\not\geq\alpha+1$.

If ${\rm RS}(\mathcal{T})\geq\alpha$ for any $\alpha$, we put ${\rm RS}(\mathcal{T})=\infty$.

A family $\mathcal{T}$ is called $e$-totally transcendental, or totally transcendental, if ${\rm RS}(\mathcal{T})$ is an ordinal.

Clearly, there are many totally transcendental families. At the same time, the following example shows that there are families which are not totally transcendental.

Example 2.1. Let $\mathcal{T}$ be a family of all theories, with infinite models, in the language $\Sigma=\{Q_{n}\mid n\in\omega\}$ of unary predicates such that any $Q_{n}$ is either empty or complete, each $T\in\mathcal{T}$ has infinitely and co-infinitely empty predicates, and each infinite a and co-infinite $\Sigma_{0}\subset\Sigma$ has a theory $T\in\mathcal{T}$ such that $Q_{n}=\emptyset$ for $T$ if and only if $Q_{n}\in\Sigma_{0}$.

Since each $\Sigma$-sentence $\varphi$ is reduced to a description of finitely many $Q_{n}$ that some of them are (non)empty, we always can divide $\mathcal{T}_{\varphi}$ into infinitely many disjoint parts with respect to some formulas. It implies that ${\rm RS}(\mathcal{T})>\alpha$ for any ordinal $\alpha$, i.e., $\mathcal{T}$ is not totally transcendental. $\Box$

By the definition, since there are ${\rm max}\{|\Sigma(\mathcal{T})|,\omega\}$ $\Sigma(\mathcal{T})$-sentences, so if ${\rm RS}(\mathcal{T})<\infty$ then $|{\rm RS}(\mathcal{T})|\leq{\rm max}\{|\Sigma(\mathcal{T})|,\omega\}$.

In particular, the following proposition holds.

Proposition 2.2. If $|\Sigma(\mathcal{T})|\leq\omega$ then either $|{\rm RS}(\mathcal{T})|\leq\omega$ or $\mathcal{T}$ is not $e$-totally transcendental.

If $\mathcal{T}$ is totally transcendental, with ${\rm RS}(\mathcal{T})=\alpha\geq 0$, we define the degree ${\rm ds}(\mathcal{T})$ of $\mathcal{T}$ as the maximal number of pairwise inconsistent sentences $\varphi_{i}$ such that ${\rm RS}(\mathcal{T}_{\varphi_{i}})=\alpha$.

Clearly, if ${\rm RS}(\mathcal{T})=\alpha$ then ${\rm ds}(\mathcal{T})\in\omega\setminus\{0\}$.

Notice also that the rank ${\rm RS}(\cdot)$ is monotone both with respect to extensions of $\mathcal{T}$ and expansions of theories in $\mathcal{T}$: if $\mathcal{T}_{1}\subseteq\mathcal{T}_{2}$ or $\mathcal{T}_{2}$ is obtained from $\mathcal{T}_{1}$ by expansions of theories in $\mathcal{T}_{1}$ then ${\rm RS}(\mathcal{T}_{1})\leq{\rm RS}(\mathcal{T}_{2})$. Besides, if ${\rm RS}(\mathcal{T}_{1})$ is an ordinal and ${\rm RS}(\mathcal{T}_{1})={\rm RS}(\mathcal{T}_{2})$ then ${\rm ds}(\mathcal{T}_{1})\leq{\rm ds}(\mathcal{T}_{2})$.

The following proposition is obvious.

Proposition 2.3. A family $\mathcal{T}$ is $e$-minimal if and only if ${\rm RS}(\mathcal{T})=1$ and ${\rm ds}(\mathcal{T})=1$.

Thus, we have an additional, with respect to Theorem 1.4, characterization of $e$-categoricity in terms of ranks.

Remark 2.4. Clearly, if ${\rm RS}(\mathcal{T})>0$ is an ordinal then $\mathcal{T}$ can be expanded obtaining a family $\mathcal{T}^{\prime}$ such that ${\rm RS}(\mathcal{T}^{\prime})>{\rm RS}(\mathcal{T})$. Indeed, each $e$-minimal subfamily $\mathcal{T}_{\varphi}$ of $\mathcal{T}$ can be divided into countably many infinite parts just introducing countably many new predicate such that these predicates are either empty or complete and for any partition of $\mathcal{T}$ into countably many infinite parts $\mathcal{T}_{i}$ each part can be labelled by a sentence that some new predicate in nonempty. This procedure increase finite rank ${\rm RS}(\mathcal{T})$ till ${\rm RS}(\mathcal{T})+1$. If ${\rm RS}(\mathcal{T})$ is infinite, we increase this rank either continuing to divide $e$-minimal $\mathcal{T}_{\varphi}$ and obtaining ${\rm RS}(\mathcal{T}^{\prime})$, or using similar expansions by new empty and complete predicates preserving $e$-minimality but increasing possibilities of other steps including limit ones and obtaining an ordinal ${\rm RS}(\mathcal{T}^{\prime})>{\rm RS}(\mathcal{T})$.

Proposition 2.5. For any infinite family $\mathcal{T}$, $e$-${\rm Sp}(\mathcal{T})$ is finite if and only if ${\rm RS}(\mathcal{T})=1$. If ${\rm RS}(\mathcal{T})=1$ then $e$-${\rm Sp}(\mathcal{T})={\rm ds}(\mathcal{T})$.

Proof. If $e$-${\rm Sp}(\mathcal{T})$ is finite then ${\rm RS}(\mathcal{T})=1$ following the proof of Proposition 1.5. Conversely, if ${\rm RS}(\mathcal{T})=1$ then $\mathcal{T}$ is divided onto ${\rm ds}(\mathcal{T})$ disjoint $e$-minimal subfamilies $\mathcal{T}_{\varphi}$, each of which has a proper accumulation point, and all accumulation points for $\mathcal{T}$ are exhausted by these ones. Thus, $e$-${\rm Sp}(\mathcal{T})={\rm ds}(\mathcal{T})$. $\Box$

Proposition 2.6. If ${\rm RS}(\mathcal{T})\geq 2$ then $e$-${\rm Sp}(\mathcal{T})\geq\omega$.

Proof. Since each infinite neighbourhood $\mathcal{T}_{\varphi}$ has an accumulation point $T$ containing $\varphi$ and by ${\rm RS}(\mathcal{T})\geq 2$ there are infinitely many disjoint infinite neighbourhoods $\mathcal{T}_{\varphi}$ we have infinitely many accumulation points each of which can be counted for the value $e$-${\rm Sp}(\mathcal{T})$. Thus, $e$-${\rm Sp}(\mathcal{T})\geq\omega$. $\Box$

The following example of a family $\mathcal{T}$ with ${\rm RS}(\mathcal{T})=2$ illustrates an existence of an accumulation point $T$ such that $T\notin{\rm Cl}_{E}(\mathcal{T}_{\varphi_{n}})$, $n\in\omega$, where the families $\mathcal{T}_{\varphi_{n}}$ divide $\mathcal{T}$ disjointly on $e$-minimal parts.

Example 2.7. Let $\mathcal{T}$ consists of theories $T^{k}_{n}$ with infinite models, in the language $\Sigma=\{Q^{k}\mid k\in\omega\}\cup\{Q^{k}_{n}\mid k,n\in\omega\}$ of unary predicates each of which is either empty or complete, where $Q^{m}$ and $Q^{m}_{s}$ are empty for $T^{k}_{n}$, if $m\neq k$, $Q^{k}$ is complete for $T^{k}_{n}$, and $Q^{k}_{m}$ is complete for $T^{k}_{n}$ if and only if $m\geq n$. The neighbourhoods $\mathcal{T}_{\varphi_{k}}$, where $\varphi_{k}$ witnesses that $Q^{k}$ is complete, divide $\mathcal{T}$ onto countably many parts, each of which is $e$-minimal. At the same time $\mathcal{T}$ has an additional accumulation point $T$, whose all unary predicates are empty. $\Box$

More generally, if there are infinitely many infinite (in particular, $e$-minimal) families $\mathcal{T}_{\varphi_{k}}$ with pairwise inconsistent sentences $\varphi_{k}$, $k\in\omega$, then $\mathcal{T}$ has an accumulation point $T$ containing $\{\neg\varphi_{k}\mid k\in\omega\}$. Indeed, each family $\mathcal{T}_{\psi_{n}}$, where $\psi_{n}=\neg\varphi_{0}\wedge\ldots\wedge\neg\varphi_{n}$, is infinite. Therefore we can construct $T$ repeating the arguments for Theorem 1.6.

Thus, we have the following:

Proposition 2.8. If there are infinite families $\mathcal{T}_{\varphi_{k}}$ with pairwise inconsistent sentences $\varphi_{k}$, $k\in\omega$, (witnessing ${\rm RS}(\mathcal{T})\geq 2$) then there is an accumulation point $T$ for $\mathcal{T}$ which is not an accumulation point for any $\mathcal{T}_{\varphi_{k}}$, $k\in\omega$.

The following modification of Example 2.7 shows that, having ${\rm RS}(\mathcal{T})=2$, the number of accumulation points in $\bigcap\limits_{k\in\omega}\mathcal{T}_{\neg\varphi_{k}}$ can vary from $1$ to $\omega$.

Example 2.9. Obtaining $n\in\omega$ additional accumulation points it suffices take the family $\mathcal{T}$ in Example 2.7 and to mark exactly one theory in each $\mathcal{T}_{\varphi_{k}}$ by some new complete predicate $R_{i}$ such that new accumulation point has exactly one complete predicate $R_{i}$. Clearly, we can mark $n$ disjoint sequences of theories producing $n$ new accumulation points. And it is possible to continue this process obtaining a family $\mathcal{T}^{\prime}$ with $\omega$ accumulation points. This process preserves $e$-minimality for $\mathcal{T}^{\prime}_{\varphi_{k}}$ and gives the values ${\rm RS}(\mathcal{T}^{\prime})=2$ and ${\rm ds}(\mathcal{T}^{\prime})=1$. $\Box$

It is easy to see that Example 2.9 can be naturally modified for an arbitrarily large language, by additional complete and empty predicates $R_{i}$ such that exactly one $R_{i}$ is complete for a chosen theory, producing a family $\mathcal{T}$ with ${\rm RS}(\mathcal{T})=2$, ${\rm ds}(\mathcal{T})=1$ and $e$-${\rm Sp}(\mathcal{T})$ equals a chosen cardinality $\lambda>\omega$.

Theorem 2.10. For any family $\mathcal{T}$, ${\rm RS}(\mathcal{T})={\rm RS}({\rm Cl}_{E}(\mathcal{T}))$, and if $\mathcal{T}$ is nonempty and $e$-totally transcendental then ${\rm ds}(\mathcal{T})={\rm ds}({\rm Cl}_{E}(\mathcal{T}))$.

Proof. At first we argue to show that ${\rm RS}(\mathcal{T})={\rm RS}({\rm Cl}_{E}(\mathcal{T}))$. Since ${\rm RS}(\mathcal{T}_{1})\leq{\rm RS}(\mathcal{T}_{2})$ for $\mathcal{T}_{1}\subseteq\mathcal{T}_{2}$, and $\mathcal{T}\subseteq{\rm Cl}_{E}(\mathcal{T})$, we have ${\rm RS}(\mathcal{T})\leq{\rm RS}({\rm Cl}_{E}(\mathcal{T}))$. Now we will prove the inequality

[TABLE]

by induction. If $\mathcal{T}$ is finite then ${\rm Cl}_{E}(\mathcal{T})=\mathcal{T}$ and the inequality (1) is obvious. If ${\rm RS}(\mathcal{T})=1$ then by Propositions 2.3 and 2.5, $\mathcal{T}$ is a finite (with ${\rm ds}(\mathcal{T})$ parts) disjoint union of $e$-minimal, i.e., $e$-categorical families $\mathcal{T}_{\varphi}$ such that ${\rm Cl}_{E}(\mathcal{T})=\bigcup\limits_{\varphi}{\rm Cl}_{E}(\mathcal{T}_{\varphi})$ and $|{\rm Cl}_{E}(\mathcal{T})\setminus\mathcal{T}|\leq{\rm ds}(\mathcal{T})<\omega$. Then ${\rm Cl}_{E}(\mathcal{T})$ is a finite disjoint union of ${\rm ds}(\mathcal{T})$ $e$-minimal families ${\rm Cl}_{E}(\mathcal{T}_{\varphi})$ producing the inequality (1), with ${\rm ds}(\mathcal{T})={\rm ds}({\rm Cl}_{E}(\mathcal{T}))$.

If ${\rm RS}({\rm Cl}_{E}(\mathcal{T}))\geq\alpha$ for a limit ordinal $\alpha$ then ${\rm RS}(\mathcal{T})\geq\alpha$ by induction. So it suffices to observe ${\rm RS}(\mathcal{T})\geq\alpha+1$ if ${\rm RS}({\rm Cl}_{E}(\mathcal{T}))\geq\alpha+1$. But if the latter inequality is witnessed by some sentences $\varphi_{n}$, $n\in\omega$, with ${\rm RS}({\rm Cl}_{E}(\mathcal{T}_{\varphi_{n}}))\geq\alpha$ then by induction ${\rm RS}(\mathcal{T}_{\varphi_{n}})\geq\alpha$, with ${\rm Cl}_{E}(\mathcal{T}_{\varphi_{n}})={\rm Cl}_{E}(\mathcal{T})_{\varphi_{n}}$. Therefore, ${\rm RS}(\mathcal{T})\geq\alpha+1$ witnessed by the same sentences $\varphi_{n}$.

Thus, ${\rm RS}(\mathcal{T})={\rm RS}({\rm Cl}_{E}(\mathcal{T}))$.

The condition ${\rm ds}(\mathcal{T})=k\Leftrightarrow{\rm ds}({\rm Cl}_{E}(\mathcal{T}))=k$ follows again by the equality ${\rm Cl}_{E}(\mathcal{T}_{\varphi_{n}})={\rm Cl}_{E}(\mathcal{T})_{\varphi_{n}}$, where the $E$-closures of disjoint neighbourhoods $\mathcal{T}_{\varphi_{n}}$, with ${\rm RS}(\mathcal{T}_{\varphi_{n}})={\rm RS}(\mathcal{T})$, ${\rm ds}(\mathcal{T}_{\varphi_{n}})=1$, exhaust ${\rm Cl}_{E}(\mathcal{T})$. $\Box$

Notice that Example 2.7 can be naturally generalized in a countable language of unary predicates producing a family $\mathcal{T}$ with given countable ordinal $\alpha={\rm RS}(\mathcal{T})$ and given positive natural number $n={\rm ds}(\mathcal{T})$. Thus, the hierarchy of families $\mathcal{T}$, in countable languages, with respect to pairs pairs $(\alpha,n)=({\rm RS}(\mathcal{T}),{\rm ds}(\mathcal{T}))$ can be realized.

If the language $\Sigma$ is uncountable we can continue the process increasing ${\rm RS}(\mathcal{T})$ to uncountable ordinals with an upper bound $|\Sigma|$, since this bound equals the cardinality of the set of all $\Sigma$-sentences $\varphi$, defining $\mathcal{T}_{\varphi}$.

Therefore the following proposition holds.

Proposition 2.11. For any ordinal $\alpha$ and a natural number $n\in\omega\setminus\{0\}$ there is a family $T$ such that $({\rm RS}(\mathcal{T}),{\rm ds}(\mathcal{T}))=(\alpha,n)$.

Having a hierarchy with $({\rm RS}(\mathcal{T}),{\rm ds}(\mathcal{T}))=(\alpha,n)$ and Proposition 2.5 for $({\rm RS}(\mathcal{T}),{\rm ds}(\mathcal{T}))=(1,n)$, it is natural to characterize these values $(\alpha,n)$ for $\alpha\geq 2$.

Definition. A family $\mathcal{T}$, with infinitely many accumulation points, is called $a$-minimal if for any sentence $\varphi\in\Sigma(T)$, $\mathcal{T}_{\varphi}$ or $\mathcal{T}_{\neg\varphi}$ has finitely many accumulation points.

The following theorem gives a characterization, in terms of $a$-minimality, for ${\rm RS}(\mathcal{T})=2$. Notice that by Theorem 2.10 it does not matter $\mathcal{T}$ is $E$-closed or not.

Theorem 2.12. For any family $\mathcal{T}$, ${\rm RS}(\mathcal{T})=2$, with ${\rm ds}(\mathcal{T})=n$, if and only if $\mathcal{T}$ is represented as a disjoint union of subfamilies $\mathcal{T}_{\varphi_{1}},\ldots,\mathcal{T}_{\varphi_{n}}$, for some pairwise inconsistent sentences $\varphi_{1},\ldots,\varphi_{n}$, such that each $\mathcal{T}_{\varphi_{i}}$ is $a$-minimal.

Proof. Let ${\rm RS}(\mathcal{T})=2$ and ${\rm ds}(\mathcal{T})=n$. By the definition $\mathcal{T}$ is represented as a disjoint union of subfamilies $\mathcal{T}_{\varphi_{1}},\ldots,\mathcal{T}_{\varphi_{n}}$, for some sentences $\varphi_{1},\ldots,\varphi_{n}$, such that each $\mathcal{T}_{\varphi_{i}}$ satisfies ${\rm RS}(\mathcal{T}_{\varphi_{i}})=2$ and ${\rm ds}(\mathcal{T}_{\varphi_{i}})=1$. So it suffices to show that, assuming ${\rm ds}(\mathcal{T})=1$, ${\rm RS}(\mathcal{T})=2$ if and only if $\mathcal{T}$ is $a$-minimal.

Let ${\rm RS}(\mathcal{T})=2$ and ${\rm ds}(\mathcal{T})=1$. Therefore have infinitely many accumulation points belonging to the $E$-closures of $e$-minimal subfamilies $\mathcal{T}_{\varphi}$. If $\mathcal{T}$ is not $a$-minimal then for some sentence $\psi\in\Sigma(T)$, $\mathcal{T}_{\psi}$ and $\mathcal{T}_{\neg\psi}$ have infinitely many accumulation points. By Proposition 2.5, ${\rm RS}\left(\mathcal{T}_{\varphi}\right)=2$ and ${\rm RS}\left(\mathcal{T}_{\neg\varphi}\right)=2$ contradicting ${\rm ds}(\mathcal{T})=1$.

Now let $\mathcal{T}$ be $a$-minimal. Having infinitely many accumulation points for $\mathcal{T}$ it is easy to construct step-by-step infinitely many disjoint infinite subfamilies $\mathcal{T}_{\psi_{i}}$, $i\in\omega$, with pairwise inconsistent sentences $\psi_{i}$, witnessing ${\rm RS}(\mathcal{T})\geq 2$. Moreover, since $\mathcal{T}$ is $a$-minimal it is possible to choose $\psi_{i}$ such that each $\mathcal{T}_{\psi_{i}}$ has unique accumulation point, i.e., by Theorem 1.4 and Proposition 2.3, it is $e$-minimal with ${\rm RS}\left(\mathcal{T}_{\psi_{i}}\right)=1$ and ${\rm ds}\left(\mathcal{T}_{\psi_{i}}\right)=1$. And each possibility to divide $\mathcal{T}$ by sentences witnessing ${\rm RS}(\mathcal{T})\geq 2$ is reduced to the case above. It means that ${\rm RS}(\mathcal{T})=2$. Since, by $a$-minimality, $\mathcal{T}$ can not be divided, by a sentence $\chi$, to subfamilies $\mathcal{T}_{\chi}$ and $\mathcal{T}_{\neg\chi}$ with infinitely many accumulation points, ${\rm ds}(\mathcal{T})=1$. $\Box$

Below we generalize the notions of $e$-minimality and $a$-minimality for arbitrary nonempty $e$-totally transcendental families $\mathcal{T}$ allowing to characterize step-by-step families of ranks $\alpha$ starting with $\alpha\in\{0,1,2\}$.

Definition. Let $\alpha$ be an ordinal. A family $\mathcal{T}$ of rank $\alpha$ is called $\alpha$-minimal if for any sentence $\varphi\in\Sigma(T)$, ${\rm RS}(\mathcal{T}_{\varphi})<\alpha$ or ${\rm RS}(\mathcal{T}_{\neg\varphi})<\alpha$.

By the definition and in view of Proposition 2.3 and Theorem 2.12 we have:

Proposition 2.13. $(1)$* A family $\mathcal{T}$ is [math]-minimal if and only if $\mathcal{T}$ is a singleton.*

$(2)$* A family $\mathcal{T}$ is $1$-minimal if and only if $\mathcal{T}$ is $e$-minimal.*

$(3)$* A family $\mathcal{T}$ is $2$-minimal if and only if $\mathcal{T}$ is $a$-minimal.*

$(4)$* For any ordinal $\alpha$ a family $\mathcal{T}$ is $\alpha$-minimal if and only if ${\rm RS}(\mathcal{T})=\alpha$ and ${\rm ds}(\mathcal{T})=1$. *

In view of Proposition 2.13 the following assertion obviously generalizes Theorem 2.12.

Proposition 2.14. For any family $\mathcal{T}$, ${\rm RS}(\mathcal{T})=\alpha$, with ${\rm ds}(\mathcal{T})=n$, if and only if $\mathcal{T}$ is represented as a disjoint union of subfamilies $\mathcal{T}_{\varphi_{1}},\ldots,\mathcal{T}_{\varphi_{n}}$, for some pairwise inconsistent sentences $\varphi_{1},\ldots,\varphi_{n}$, such that each $\mathcal{T}_{\varphi_{i}}$ is $\alpha$-minimal.

3 Boolean algebras and CB-ranks

Similarly [8], for a nonempty family $\mathcal{T}$, we denote by $\mathcal{B}(\mathcal{T})$ the Boolean algebra consisting of all subfamilies $\mathcal{T}_{\varphi}$, where $\varphi$ are sentences in the language $\Sigma(\mathcal{T})$.

Following [8] we observe that $\mathcal{B}(\mathcal{T})$ is superatomic [10, 11] for every $e$-totally transcendental $\mathcal{T}$, with well-ordered chains. And vice versa, having superatomic $\mathcal{B}(\mathcal{T})$ we step-by-step define ordinals ${\rm RS}(\mathcal{T}_{\varphi})$ for $\mathcal{T}_{\varphi}$ implying that $\mathcal{T}$ is $e$-totally transcendental. Thus, the following theorem holds.

Theorem 3.1. A nonempty family $\mathcal{T}$ is $e$-totally transcendental if and only if the Boolean algebra $\mathcal{B}(\mathcal{T})$ is superatomic.

In particular, for an infinite family $\mathcal{T}$, the start of the process, producing an ordinal ${\rm RS}(\mathcal{T})$, should be bases on $e$-minimal families $\mathcal{T}_{\varphi}$, i.e., if each infinite $\mathcal{T}_{\varphi}$ is definably divided into two infinite parts $\mathcal{T}_{\varphi\wedge\psi}$ and $\mathcal{T}_{\varphi\wedge\neg\psi}$, then $\mathcal{T}_{\varphi}$, and, in particular, $\mathcal{T}=\mathcal{T}_{\forall x(x\approx x)}$, has ${\rm RS}(\mathcal{T}_{\varphi})=\infty$.

Thus we have the following

Proposition 3.2. If an infinite family $\mathcal{T}$ does not have $e$-minimal subfamilies $\mathcal{T}_{\varphi}$ then $\mathcal{T}$ is not $e$-totally transcendental.

Remark 3.3. By the definition of the rank, for any family $\mathcal{T}$ represented as a union $\mathcal{T}_{1}\cup\mathcal{T}_{2}$ we have ${\rm RS}(\mathcal{T})={\rm max}\{{\rm RS}(\mathcal{T}_{1}),{\rm RS}(\mathcal{T}_{2})\}$ since each step for ${\rm RS}(\mathcal{T})$ uses infinitely many theories in $\mathcal{T}_{1}$ or $\mathcal{T}_{2}$ dividing some neighbourhoods $(\mathcal{T}_{i})_{\varphi}$ into infinitely many disjoint parts. At the same time, ${\rm ds}(\mathcal{T})$ can vary from ${\rm max}\{{\rm ds}(\mathcal{T}_{1}),{\rm ds}(\mathcal{T}_{2})\}$ till ${\rm ds}(\mathcal{T}_{1})+{\rm ds}(\mathcal{T}_{2})$ depending on $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$. $\Box$

Recall the definition of the Cantor–Bendixson rank. It is defined on the elements of a topological space $X$ by induction: ${\rm CB}_{X}(p)\geq 0$ for all $p\in X$; ${\rm CB}_{X}(p)\geq\alpha$ if and only if for any $\beta<\alpha$, $p$ is an accumulation point of the points of ${\rm CB}_{X}$-rank at least $\beta$. ${\rm CB}_{X}(p)=\alpha$ if and only if both ${\rm CB}_{X}(p)\geq\alpha$ and ${\rm CB}_{X}(p)\ngeq\alpha+1$ hold; if such an ordinal $\alpha$ does not exist then ${\rm CB}_{X}(p)=\infty$. Isolated points of $X$ are precisely those having rank [math], points of rank $1$ are those which are isolated in the subspace of all non-isolated points, and so on. For a non-empty $C\subseteq X$ we define ${\rm CB}_{X}(C)=\sup\{{\rm CB}_{X}(p)\mid p\in C\}$; in this way ${\rm CB}_{X}(X)$ is defined and ${\rm CB}_{X}(\{p\})={\rm CB}_{X}(p)$ holds. If $X$ is compact and $C$ is closed in $X$ then the sup is achieved: ${\rm CB}_{X}(C)$ is the maximum value of ${\rm CB}_{X}(p)$ for $p\in C$; there are finitely many points of maximum rank in $C$ and the number of such points is the ${\rm CB}_{X}$-degree of $C$, denoted by $n_{X}(C)$.

If $X$ is countable and compact then ${\rm CB}_{X}(X)$ is a countable ordinal and every closed subset has ordinal-valued rank and finite ${\rm CB}_{X}$-degree $n_{X}(X)\in\omega\setminus\{0\}$.

For any ordinal $\alpha$ the set $\{p\in X\mid{\rm CB}_{X}(p)\geq\alpha\}$ is called the $\alpha$-th ${\rm CB}$-derivative $X_{\alpha}$ of $X$.

Elements $p\in X$ with ${\rm CB}_{X}(p)=\infty$ form the perfect kernel $X_{\infty}$ of $X$.

Clearly, $X_{\alpha}\supseteq X_{\alpha+1}$, $\alpha\in{\rm Ord}$, and $X_{\infty}=\bigcap\limits_{\alpha\in{\rm Ord}}X_{\alpha}$.

Similarly, for a nontrivial superatomic Boolean algebra $\mathcal{A}$ the characteristics ${\rm CB}_{\mathcal{A}}(A)$, $n_{\mathcal{A}}(A)$, and ${\rm CB}_{\mathcal{A}}(p)$, for $p\in A$, are defined [11] starting with atomic elements being isolated points. Following [11], ${\rm CB}_{\mathcal{A}}(A)$ and $n_{\mathcal{A}}(A)$ are called the Cantor–Bendixson invariants, or ${\rm CB}$-invariants of $\mathcal{A}$.

Recall that by [11, Lemma 17.9], ${\rm CB}_{\mathcal{A}}(A)<|A|^{+}$ for any infinite $\mathcal{A}$, and the following theorem holds.

Theorem 3.4 [11, Theorem 17.11]. Countable superatomic Boolean algebras are isomorphic if and only if they have the same ${\rm CB}$-invariants.

In view of Theorem 3.1 any $e$-totally transcendental family $\mathcal{T}$ defines a superatomic Boolean algebra $\mathcal{B}(\mathcal{T})$, and it is easy to observe step-by-step that ${\rm RS}(\mathcal{T})={\rm CB}_{\mathcal{B}(\mathcal{T})}(B(\mathcal{T}))$, ${\rm ds}(\mathcal{T})=n_{\mathcal{B}(\mathcal{T})}(B(\mathcal{T}))$, i.e., the pair $({\rm RS}(\mathcal{T}),{\rm ds}(\mathcal{T}))$ consists of ${\rm CB}$-invariants for $\mathcal{B}(\mathcal{T})$.

In particular, by Theorem 3.4, for any countable $e$-totally transcendental family $\mathcal{T}$, $\mathcal{B}(\mathcal{T})$ is uniquely defined, up to isomorphism, by the pair $({\rm RS}(\mathcal{T}),{\rm ds}(\mathcal{T}))$ of ${\rm CB}$-invariants.

By the definition for any $e$-totally transcendental family $\mathcal{T}$ each theory $T\in\mathcal{T}$ obtains the ${\rm CB}$-rank ${\rm CB}_{\mathcal{T}}(T)$ starting with $\mathcal{T}$-isolated points $T_{0}$, of ${\rm CB}_{\mathcal{T}}(T_{0})=0$. We will denote the values ${\rm CB}_{\mathcal{T}}(T)$ by ${\rm RS}_{\mathcal{T}}(T)$ as the rank for the point $T$ in the topological space on $\mathcal{T}$ which is defined with respect to $\Sigma(\mathcal{T})$-sentences.

Remark 3.5. By the definition we have ${\rm RS}_{\mathcal{T}}(T)\geq 1$ if and only if $T$ is an accumulation point for $\mathcal{T}$, ${\rm RS}_{\mathcal{T}}(T)\geq 2$ if and only if $T$ is an accumulation point for the subfamily of ${\rm Cl}_{E}(\mathcal{T})$ consisting of all its accumulation points, etc. Additionally, by Proposition 2.14, if $\mathcal{T}$ is $E$-closed with ${\rm RS}(\mathcal{T})=\alpha\geq 0$ then $\mathcal{T}$ contains exactly ${\rm ds}(\mathcal{T})$ theories $T$ such that ${\rm RS}_{\mathcal{T}}(T)=\alpha$. If means that $\mathcal{T}$ is represented as a disjoint union of ${\rm ds}(\mathcal{T})$ $\alpha$-minimal subfamilies $\mathcal{T}_{\varphi_{i}}$ each of which has unique theory $T_{i}$ with ${\rm RS}_{\mathcal{T}}(T_{i})=\alpha$. $\Box$

4 Ranks for countable languages

Below we prove a characterization for bounds of the hierarchy of ${\rm RS}(\mathcal{T})$, for countable languages, i.e., rank bounds for $e$-totally transcendental families.

Proposition 4.1. If ${\rm RS}(\mathcal{T})=\infty$ then $|{\rm Cl}_{E}(\mathcal{T})|\geq 2^{\omega}$.

Proof. Since ${\rm RS}(\mathcal{T})=\infty$ there is a $2$-tree of sentences $\varphi_{\Delta}$, $\Delta\in{\,}^{<\omega}2$, such that $\mathcal{T}_{\varphi_{\Delta}}$ are infinite, $\varphi_{{\Delta}\,\hat{\,}\,i}\vdash\varphi_{\Delta}$, $i\in\{0,1\}$, and $\varphi_{{\Delta}\,\hat{\,}\,0}$, $\varphi_{{\Delta}\,\hat{\,}\,1}$ are inconsistent. It easy to see that for each $f\in 2^{\omega}$ there is an accumulation point $T_{f}$ for $\mathcal{T}$ containing the sentences $\varphi_{\langle f(0),\ldots,f(n)\rangle}$, $n\in\omega$. Clearly, $T_{f_{1}}\neq T_{f_{2}}$ for $f_{1}\neq f_{2}$. Hence, $|{\rm Cl}_{E}(\mathcal{T})|\geq 2^{\omega}$. $\Box$

Remark 4.2. If the language $\Sigma(\mathcal{T})$ is at most countable then the 2-tree of sentences $\varphi_{\Delta}$ in the proof of Proposition 4.1 allows to form a countable subfamily $\mathcal{T}^{\prime}$ of $\mathcal{T}$ with $e$-${\rm Sp}(\mathcal{T})=2^{\omega}$. For this aim it suffices to choose for $\mathcal{T}^{\prime\prime}$ some theories in $\mathcal{T}_{\varphi_{\Delta}}$ which do not belong to some $\mathcal{T}_{\varphi_{\Delta^{\prime}}}$, where $\Delta^{\prime}$ is a continuation of $\Delta$. The theories $T_{f}$ belong to the $E$-closure of $\mathcal{T}^{\prime}$ being the union of $\mathcal{T}^{\prime\prime}$ with some at most countable subset $\mathcal{T}^{\prime\prime\prime}$ of $\mathcal{T}$ such that each sentence $\varphi$ in any $T_{f}$ has countable $\mathcal{T}^{\prime\prime\prime}_{\varphi}$. Thus, $e$-${\rm Sp}(\mathcal{T})\geq 2^{\omega}$.

In general case, for $|\Sigma(\mathcal{T})|\leq\omega$, both infinite families $\mathcal{T}$ and ${\rm Cl}_{E}(\mathcal{T})$ are countably generated, i.e., contain a countable $\mathcal{T}^{\prime\prime\prime}$ generating both $\mathcal{T}$ and ${\rm Cl}_{E}(\mathcal{T})$. Indeed, since there are countably many $\Sigma(\mathcal{T})$-sentences $\varphi$, by Proposition 1.1 if suffices to form $\mathcal{T}^{\prime\prime\prime}$ by all finite $\mathcal{T}$-definable families $\mathcal{T}_{\varphi}$, and by arbitrary countable subfamilies of $\mathcal{T}_{\varphi}$, if $\mathcal{T}_{\varphi}$ is infinite.

Thus, for any at most countable language $\Sigma$, if $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$ then $e$-${\rm Sp}(\mathcal{T})=2^{\omega}$. Since $e$-${\rm Sp}(\mathcal{T})=2^{\omega}$ obviously implies $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$ for $|\Sigma(\mathcal{T})|\leq\omega$, we have the following:

Proposition 4.3. If $|\Sigma(\mathcal{T})|\leq\omega$ then $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$ if and only if $e$-${\rm Sp}(\mathcal{T})=2^{\omega}$.

Proposition 4.4. If $|\Sigma(\mathcal{T})|\leq\omega$ and $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$ then ${\rm RS}(\mathcal{T})=\infty$.

Proof. By Theorem 2.10 it suffices to assume that $\mathcal{T}$ is $E$-closed such that $|\mathcal{T}|=2^{\omega}$.

At first we note that there is a sentence $\varphi$ such that $|\mathcal{T}_{\varphi}|=2^{\omega}$ and $|\mathcal{T}_{\neg\varphi}|=2^{\omega}$. Indeed, assuming that $\varphi$ does not exist we can enumerate all $\Sigma(\mathcal{T})$ sentences: $\varphi_{n}$, $n\in\omega$, and form a sequence $\psi_{n}$ such that $\psi_{0}=\varphi_{0}$, $\psi_{n+1}=\psi_{n}\wedge\varphi^{\delta}_{n+1}$, $\delta\in\{0,1\}$, with $|\mathcal{T}_{\psi_{n+1}}|=2^{\omega}$. Thus, $\left|\bigcap\limits_{n=0}^{\infty}\mathcal{T}_{\psi_{n}}\right|=2^{\omega}$ contradicting the condition that $\{\psi_{n}\mid n\in\omega\}$ forces a complete theory.

Repeating the arguments we construct a $2$-tree ${\rm Tr}$ of sentences $\varphi_{\Delta}$, $\Delta\in{\,}^{<\omega}2$, as in the proof of Proposition 4.1 such that each $\varphi_{\Delta}$ satisfies $|\mathcal{T}_{\varphi_{\Delta}}|=2^{\omega}$.

Now the sentences in the $2$-tree ${\rm Tr}$ witness that $\mathcal{T}$ in not $e$-totally transcendental. Indeed, $\mathcal{T}_{\varphi_{0}}$, $\mathcal{T}_{\varphi_{10}}$, $\mathcal{T}_{\varphi_{110}},\ldots$ are disjoint families each of which has continuum many theories. Each family $\mathcal{T}_{\varphi_{1\ldots 10}}$ contains again infinitely many disjoint subfamilies $\mathcal{T}_{\varphi_{1\ldots 101\ldots 10}}$ each of which has continuum many theories. Continuing the process we observe that each $\mathcal{T}_{\varphi_{1\ldots 10}}$, and, thus, $\mathcal{T}$ have the ranks equal to $\infty$. $\Box$

Collecting Propositions 4.1, 4.3, and 4.4 we obtain:

Theorem 4.5. For any family $\mathcal{T}$ with $|\Sigma(\mathcal{T})|\leq\omega$ the following conditions are equivalent:

$(1)$* $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$;*

$(2)$* $e$-${\rm Sp}(\mathcal{T})=2^{\omega}$;*

$(3)$* ${\rm RS}(\mathcal{T})=\infty$.*

Remark 4.6. Having characterizations for $e$-totally transcendental families $\mathcal{T}$ of theories by Theorem 4.5 we observe that both theories $T$ in $e$-totally transcendental $\mathcal{T}$ can be not totally transcendental themselves, containing, for instance, countably many independent unary predicates, and totally transcendental theories, with either empty or complete predicates $Q$, as in Example 2.7, can form families $\mathcal{T}$ which are not $e$-totally transcendental, just dividing $\mathcal{T}$ by sentences describing that the predicates $Q$ are empty or complete. Thus, the notions of totally transcendental theories and $e$-totally transcendental families do not correlate in general case.

Remark 4.7. Examples in [3] show that families $\mathcal{T}$ with $|\Sigma(\mathcal{T})|\leq\omega$ and $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$ can (do not) have least generating sets. Moreover, modifications of this examples can produce families of theories with proper derivatives for arbitrary ordinals $\alpha$. Therefore the perfect kernel for $\mathcal{T}$ can be formed on some derivative step $\alpha$. Thus, for any ordinal $\alpha>0$ there is a family $\mathcal{T}$ such that $\mathcal{T}_{\alpha}=\mathcal{T}_{\infty}$ whereas $\mathcal{T}_{\beta}\neq\mathcal{T}_{\infty}$ for $\beta<\alpha$.

Remark 4.8. Notice that Theorem 4.5 does not hold for $|\Sigma(\mathcal{T})|>\omega$, in general case. Indeed, language uniform theories [3] can have both big cardinalities for languages, big cardinalities for $\mathcal{T}$ and small cardinalities for $e$-spectra. For instance, taking a family $\mathcal{T}=\{T_{i}\mid i\in I\}$ in a language $\Sigma$ of unary predicates $Q_{i}$, $i\in I$, $|I|=\lambda>\omega$, such that $T_{i}$ has complete predicate $Q_{i}$ and empty predicates $Q_{j}$, $j\neq i$, we have $|{\rm Cl}_{E}(\mathcal{T})|=\lambda$ with ${\rm Cl}_{E}(\mathcal{T})=\mathcal{T}\cup\{T_{\infty}\}$, where $T_{\infty}$ has only empty predicates, whereas $e$-${\rm Sp}(\mathcal{T})=1$, that witnessed by $T_{\infty}$. Besides, $\mathcal{T}$ is $e$-minimal, i.e., ${\rm RS}(\mathcal{T})=1$ and ${\rm ds}(\mathcal{T})=1$. In particular, for $\lambda=2^{\omega}$, we have $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$, $e$-${\rm Sp}(\mathcal{T})=1$, and ${\rm RS}(\mathcal{T})=1$ refuting Theorem 4.5 for $|\Sigma(\mathcal{T})|=2^{\omega}$.

Additionally, the family $\mathcal{T}$ can be expanded by unary disjoint predicates $Q^{\prime}_{j}$, $j\in J$, $|J|\geq 2$, such that each $T_{i}$ is extended to $T_{ij}$ obtaining complete $Q^{\prime}_{j}$ and empty $Q^{\prime}_{k}$ for $k\neq j$. The families $\mathcal{T}_{j}=\{T_{ij}\mid i\in I\}$ stay $e$-minimal, producing unique accumulation points, whereas we have for $\mathcal{T}^{\prime}=\{T_{ij}\mid i\in I,j\in J\}$:

1. if $J$ is finite then ${\rm Cl}_{E}(\mathcal{T}^{\prime})=\bigcup\limits_{j\in J}{\rm Cl}_{E}(\mathcal{T}_{j})$ and $|{\rm Cl}_{E}(\mathcal{T}^{\prime})|=|J|$; if $J$ is infinite then ${\rm Cl}_{E}(\mathcal{T}^{\prime})$ consists of $\bigcup\limits_{j\in J}{\rm Cl}_{E}(\mathcal{T}_{j})$ and $|I|$ theories with unique nonempty $Q_{i}$ and all empty $Q^{\prime}_{j}$, as well as of unique theory $T_{\infty}$ with all empty predicates; therefore $|{\rm Cl}_{E}(\mathcal{T}^{\prime})|=|I|+|J|+1=|I|+|J|$;

2. by the previous item, $e$-${\rm Sp}(\mathcal{T})=|J|$ for finite $J$, and $e$-${\rm Sp}(\mathcal{T})=|I|+|J|$ for infinite $J$;

3. ${\rm RS}(\mathcal{T})=2$.

In conclusion we formulate the following:

Problem. Describe the rank ${\rm RS}(\cdot)$ hierarchy for natural families of theories.