\documentclass[10pt]{amsart}
\usepackage[cp1251]{inputenc}
\usepackage[english,russian]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}

\def\udcs{510.67} %Here the author places classificators of the paper according to Russian classification system
\def\mscs{03C07, 03C10, 03C68} %Here the author places  classificators according to the AMS classification list.
\setcounter{page}{144}

\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}
\newtheorem{fact}{Fact}
\newtheorem{note}{Remark}
\newtheorem{example}{Example}

\def\logo{{\bf\huge S\raisebox{0.2ex}{\hspace{0.55ex}\raisebox{0.05ex}e\hspace{-1.65ex}$\bigcirc$}MR}}

\def\semrtop
     {
  \vbox{
     \noindent\logo
     \hspace{80mm}\raisebox{1ex}{ISSN 1813-3304 }

     \vspace{5mm}

     \begin{center}
     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
%     {\small Sobolev Institute of Mathematics SB RAS}
     \end{center}
     \vspace{-3mm}
     \noindent
     \begin{tabular}{c}
     \hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa} \\
     \hline\hline
     \end{tabular}

     \vspace{1mm}
     {\flushleft\it Том 19, стр. ???--??? (2022) \hspace{64mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
                                                                                    %filled by Editorial board
     \newline
         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
  }%\vbox
}

\begin{document}
\renewcommand{\refname}{References}
\renewcommand{\proofname}{Proof.}
\renewcommand{\figurename}{Fig.}


\thispagestyle{empty}

\title[Special relations for formulae, their equivalence relations]{Special relations for formulae, their equivalence
relations and theories}
\author{S.V. Sudoplatov}%



\address{Sergey Vladimirovich Sudoplatov
\newline\hphantom{iii} Sobolev Institute of Mathematics
\newline\hphantom{iii} Academician Koptyug avenue, 4
\newline\hphantom{iii} 630090, Novosibirsk, Russia.
}%
\email{sudoplat@math.nsc.ru}
\address{Novosibirsk
State Technical University
\newline\hphantom{iii} K. Marx avenue, 20
\newline\hphantom{iii} 630073, Novosibirsk, Russia.
}%
\email{sudoplatov@corp.nstu.ru}%


\thanks{\sc Sudoplatov, S.V.,
Special relations for formulae, their equivalence relations and
theories}
\thanks{\copyright \ 2022 Sudoplatov S.V}
\thanks{\rm The work was carried out in the framework of the
State Contract of the Sobolev Institute of Mathematics, Project
No.~FWNF-2022-0012, of the Committee of Science in Education and
the Science Ministry of the Republic of Kazakhstan, Grant No.
AP08855497 (Sections 2 and 3), and of Russian Scientific
Foundation, Project No.~22-21-00044 (Sections 4 and 5).}
\thanks{\it Received  ???, ??, 2022, published  ???, ??,  2022.}%

\semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} We study properties and characteristics
for special relations of formulae in given languages producing
equivalence relations both in general case and for elementary
theories. Characteristics for chains of special relations and
equivalence relations are described.\medskip

\noindent{\bf Keywords:} formula, special relation, equivalence
relation, theory.
 \end{quote}
}

\bigskip
Studying valuable links between theories structural relations were
introduced both for theories in given languages and for theories
in distinct languages including their similarity \cite{Peret},
definability \cite{Er_def}, interpretability \cite{PiG}, etc. We
introduced and studied properties and ranks for families of
theories in a given language \cite{RSrank, MarkhSud, dst, M19,
MS19, Markh, SudFP, KS22}.

In the present paper we introduce and study both special relations
for formulae and equivalence ones both in general case and for
elementary theories. Characteristics for chains of special
relations and equivalence relations are described. These
con\-si\-de\-ra\-t\-ions are based both on transformations of the
rank ${\rm RS}$ \cite{RSrank} as well as on a general approach for
a hierarchy of links between definable relations \cite{MT}.

The paper is arranged as follows. Preliminary notions, notations
and assertions are represented in Section 1. In Section 2, we
consider special and equivalence relations on sets of formulae,
their forms and properties. Elementary special and equivalence
relations are studies in Section 3, including categories related
to these relations. Complete and precomplete special and
equivalence relations, and their completions, possibilities for
ranks are described in Section 4. In Section 5 chains of special
and equivalence relations are studied. Spectra for finite lengths
of chains are characterized and spectra for the lengths with
respect to countable languages are described.

\section{Preliminaries}

Following \cite{RSrank} we define the {\em rank} ${\rm RS}(\cdot)$
for families $\mathcal{T}$ of first-order theories in a language
$\Sigma=\Sigma(\mathcal{T})$, similar to Morley rank for a fixed
theory, and a hierarchy with respect to these ranks in the
following way.

By $F(\Sigma)$ we denote the set of all formulae in the language
$\Sigma$ and by ${\rm Sent}(\Sigma)$ the set of all sentences in
$F(\Sigma)$.

For a sentence $\varphi\in {\rm Sent}(\Sigma)$ we denote by
$\mathcal{T}_\varphi$ the set of all theories $T\in\mathcal{T}$
with $\varphi\in T$.

Any set $\mathcal{T}_\varphi$ is called the {\em
$\varphi$-neighbourhood}, or simply a {\em neighbourhood}, for
$\mathcal{T}$, or the ($\varphi$-){\em definable} subset of
$\mathcal{T}$. The set $\mathcal{T}_\varphi$ is also called ({\em
formula-} or {\em sentence-}){\em definable} (by the sentence
$\varphi$) with respect to $\mathcal{T}$, or ({\em
sentence-})$\mathcal{T}$-{\em definable}, or simply {\em
$s$-definable}.

\medskip
{\bf Definition} \cite{dst}. If $\mathcal{T}$ is a family of
theories and $\Phi$ is a set of sentences, then we put
$\mathcal{T}_\Phi=\bigcap\limits_{\varphi\in\Phi}\mathcal{T}_\varphi$
and the set $\mathcal{T}_\Phi$ is called ({\em type-} or {\em
diagram-}){\em definable} (by the set $\Phi$) with respect to
$\mathcal{T}$, or ({\em diagram-})$\mathcal{T}$-{\em definable},
or simply {\em $d$-definable}.

Clearly, finite unions of $d$-definable sets are again
$d$-definable. Considering infinite unions $\mathcal{T}'$ of
$d$-definable sets $\mathcal{T}_{\Phi_i}$, $i\in I$, one can
represent them by sets of sentences with infinite disjunctions
$\bigvee\limits_{i\in I}\varphi_i$, $\varphi_i\in\Phi_i$. We call
these unions $\mathcal{T}'$ are called {\em $d_\infty$-definable}
sets.

\medskip
{\bf Definition} \cite{RSrank}. For the empty family $\mathcal{T}$
we put the rank ${\rm RS}(\mathcal{T})=-1$, and for nonempty
families $\mathcal{T}$ we put ${\rm RS}(\mathcal{T})\geq 0$.

For a family $\mathcal{T}$ and an ordinal $\alpha=\beta+1$ we put
${\rm RS}(\mathcal{T})\geq\alpha$ if there are pairwise
$\mathcal{T}$-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 {\em $e$-totally transcendental},
or {\em totally transcendental}, if ${\rm RS}(\mathcal{T})$ is an
ordinal.

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

\medskip
As noticed in \cite{MS_top1, MS_top2} these notions are valid both
for families of complete theories and for arbitrary families of
theories including incomplete ones.

Below in this section we consider results on families of complete
theories in a given language.

The following theorem characterizes the property of $e$-total
transcendency for countable languages.

\medskip
{\bf Theorem 1.1} \cite{RSrank, fpfag}. {\em For any family
$\mathcal{T}$ with $|\Sigma(\mathcal{T})|\leq\omega$ the following
conditions are equivalent:

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

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

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

$(4)$ there exists a $2$-tree of sentences $\varphi$ for
$s$-definable sets $\mathcal{T}_\varphi$.}

\medskip
Let $\Sigma$ be a language. If $\Sigma$ is relational we denote by
$\mathcal{T}_\Sigma$ the family of all theories of the language
$\Sigma$. If $\Sigma$ contains functional symbols $f$ then
$\mathcal{T}_\Sigma$ is the family of all theories of the language
$\Sigma'$, which is obtained by replacements of all $n$-ary
symbols $f$ with $(n+1)$-ary predicate symbols $R_f$ which
interpreted by $R_f=\{(\bar{a},b)\mid f(\bar{a})=b\}$.

\medskip
{\bf Theorem 1.2} \cite{MarkhSud}. {\em For any language $\Sigma$
either ${\rm RS}(\mathcal{T}_\Sigma)$ is finite, if $\Sigma$
consists of finitely many $0$-ary and unary predicates, and
finitely many constant symbols, or ${\rm
RS}(\mathcal{T}_\Sigma)=\infty$, otherwise.}

\medskip
For a language $\Sigma$ we denote by $\mathcal{T}_{\Sigma,n}$ the
family of all theories in $\mathcal{T}_\Sigma$ having $n$-element
models, $n\in\omega$, as well as by $\mathcal{T}_{\Sigma,\infty}$
the family of all theories in $\mathcal{T}_\Sigma$ having infinite
models.

\medskip
{\bf Theorem 1.3} \cite{MarkhSud}. {\em For any language $\Sigma$
either ${\rm RS}(\mathcal{T}_{\Sigma,n})=0$, if $\Sigma$ is finite
or $n=1$ and $\Sigma$ has finitely many predicate symbols, or
${\rm RS}(\mathcal{T}_{\Sigma,n})=\infty$, otherwise.}

\medskip
{\bf Theorem 1.4} \cite{MarkhSud}. {\em For any language $\Sigma$
either ${\rm RS}(\mathcal{T}_{\Sigma,\infty})$ is finite, if
$\Sigma$ is finite and without predicate symbols of arities $m\geq
2$ as well as without functional symbols of arities $n\geq 1$, or
${\rm RS}(\mathcal{T}_{\Sigma,\infty})=\infty$, otherwise.}

\medskip
By the definition the families $\mathcal{T}_\Sigma$,
$\mathcal{T}_{\Sigma,n}$, $\mathcal{T}_{\Sigma,\infty}$ are
$E$-closed. Thus, combining Theorem 1.1 with Theorems 1.2--1.4 we
obtain the following possibilities of car\-di\-na\-li\-t\-ies for
the families $\mathcal{T}_\Sigma$, $\mathcal{T}_{\Sigma,n}$,
$\mathcal{T}_{\Sigma,\infty}$ depending on $\Sigma$ and
$n\in\omega$:

\medskip
{\bf Proposition 1.5.} {\em For any language $\Sigma$ either
either $\mathcal{T}_\Sigma$ is countable, if if $\Sigma$ consists
of finitely many $0$-ary and unary predicates, and finitely many
constant symbols, or $|\mathcal{T}_\Sigma|\geq 2^\omega$,
otherwise.}

\medskip
{\bf Proposition 1.6.} {\em For any language $\Sigma$ either
$\mathcal{T}_{\Sigma,n}$ is finite, if $\Sigma$ is finite or $n=1$
and $\Sigma$ has finitely many predicate symbols, or
$|\mathcal{T}_{\Sigma,n}|\geq 2^\omega$, otherwise.}

\medskip
{\bf Proposition 1.7.} {\em For any language $\Sigma$ either
$\mathcal{T}_{\Sigma,\infty}$ is at most countable, if $\Sigma$ is
finite and without predicate symbols of arities $m\geq 2$ as well
as without functional symbols of arities $n\geq 1$, or
$|\mathcal{T}_{\Sigma,\infty}|\geq 2^\omega$, otherwise.}

\medskip
{\bf Definition} \cite{dst}. Let $\mathcal{T}$ be a family of
theories, $\Phi$ be a set of sentences, $\alpha$ be an ordinal
$\leq{\rm RS}(\mathcal{T})$ or $-1$. The set $\Phi$ is called {\em
$\alpha$-ranking} for $\mathcal{T}$ if ${\rm
RS}(\mathcal{T}_\Phi)=\alpha$. A sentence $\varphi$ is called {\em
$\alpha$-ranking} for $\mathcal{T}$ if $\{\varphi\}$ is {\em
$\alpha$-ranking} for $\mathcal{T}$.

The set $\Phi$ (the sentence $\varphi$) is called {\em ranking}
for $\mathcal{T}$ if it is $\alpha$-ranking for $\mathcal{T}$ with
some $\alpha$.

\medskip
{\bf Proposition 1.8} \cite{dst}. {\em For any ordinals
$\alpha\leq\beta$, if ${\rm RS}(\mathcal{T})=\beta$ then ${\rm
RS}(\mathcal{T}_\varphi)=\alpha$ for some {\rm
(}$\alpha$-ranking{\rm )} sentence $\varphi$. Moreover, there are
${\rm ds}(\mathcal{T})$ pairwise $\mathcal{T}$-inconsistent
$\beta$-ranking sentences for $\mathcal{T}$, and if $\alpha<\beta$
then there are infinitely many pairwise $\mathcal{T}$-inconsistent
$\alpha$-ranking sentences for $\mathcal{T}$.}

\medskip
{\bf Theorem 1.9} \cite{dst}. {\em Let $\mathcal{T}$ be a family
of a countable language $\Sigma$ and with ${\rm
RS}(\mathcal{T})=\infty$, $\alpha$ be a countable ordinal,
$n\in\omega\setminus\{0\}$. Then there is a $d_\infty$-definable
subfamily $\mathcal{T}^\ast\subset\mathcal{T}$ such that ${\rm
RS}(\mathcal{T}^\ast)=\alpha$ and ${\rm ds}(\mathcal{T}^\ast)=n$.}


\section{Special and equivalence relations on sets of formulae}

{\bf Definition.} Let $F(\Sigma)$ be the set of all formulae in a
language $\Sigma$, $V$ be an infinite set of variables, $V^\ast$
be the set of all tuples $\overline{x}\in V^n$, $n\in\omega$. A
ternary relation $E\subseteq F(\Sigma)\times F(\Sigma)\times
V^\ast$ is called {\em special} (for $F(\Sigma)$) if for each
$\overline{x}\in V^\ast$,
$$E_{\overline{x}}=\{(\varphi,\psi)\mid (\varphi,\psi,\overline{x})\in
E\}$$ is an equivalence relation on the set $X\subset F(\Sigma)$
consisting of all formulae $\varphi$ whose each free variable
belongs to $\overline{x}$.

\medskip
Since each formula $\varphi=\varphi(\overline{x})\in F(\Sigma)$
can be considered as a formula $\varphi(\overline{y})$ for any
$\overline{y}\supseteq\overline{x}$, then for any special relation
$E$ for $F(\Sigma)$, $\varphi$ belongs to some
$E_{\overline{y}}$-class for any
$\overline{y}\supseteq\overline{x}$. Therefore for any $\varphi\in
F(\Sigma)$ there are infinitely many equivalence classes for $E$,
within distinct $\overline{y}\in V^\ast$, containing $\varphi$.

\medskip
{\bf Definition.} The special relation $E\in{\rm SR}(\Sigma)$ is
called {\em coordinated} if the relation
$$E^\ast=\{(\varphi,\psi)\mid (\varphi,\psi,\overline{x})\in
E\mbox{ for some }\overline{x}\}$$ is an equivalence relation on
$F(\Sigma)$.

We denote by ${\rm id}_F(\Sigma)$ the special relation $E$ for
$F(\Sigma)$ whose all $E_{\overline{x}}$-classes are singletons.

\medskip
By the definition ${\rm id}_F(\Sigma)$ is the least special
relation for $F(\Sigma)$.

Clearly, there are both coordinated special relations (for
instance, ${\rm id}_F(\Sigma)$) and non-coordinated ones since
equivalence classes for $E_{\overline{x}}$ can be arbitrarily
varied extending $\overline{x}$. Notice also that by the
definition the restriction of $E^\ast$, for coordinated $E$, to
the set of all sentences in $F(\Sigma)$ is an equivalence
relation, which both can be equal to $E_{\emptyset}$ or strictly
contain $E_{\emptyset}$.

\medskip
{\bf Definition.} A special relation $E$ is called {\em upward
directed} if $E_{\overline{x}}\cup E_{\overline{y}}\subseteq
E_{\overline{x}\,\hat{\,}\,\overline{y}}$ for any
$\overline{x},\overline{y}\in V^\ast$.

\medskip
By the definition each upward directed special relation is
coordinated.

\medskip
For each formula $\varphi=\varphi(\overline{x})\in F(\Sigma)$ and
a special relation $E$ we consider the set $\varphi_E=\{\forall
\overline{x}
(\varphi(\overline{x})\leftrightarrow\psi(\overline{x}))\mid(\varphi,\psi,\overline{x})\in
E\}$. We denote by $\nabla_E$ the set $\bigcup\limits_{\varphi\in
F(\Sigma)}\varphi_E$.

\medskip
{\bf Example 2.1.} Let $\varphi(\overline{x})$ be a formula in a
language $\Sigma$, $R$ be a new relational symbol of arity
$l(\overline{x})$, $\Sigma'$ be an expansion of $\Sigma$
containing $R$. If $E$ is the equivalence relation on $F(\Sigma')$
with one two-element class
$\{\varphi(\overline{x}),R(\overline{x})\}$ and singletons for all
other formulae we obtain $\nabla_E$ generated by the {\em
$\varphi$-naming}, or the naming for $\varphi(\overline{x})$:
$\forall \overline{x} (\varphi(\overline{x})\leftrightarrow
R(\overline{x}))$.

If $\nabla_E$ is generated by $\varphi$-namings for each
$\varphi\in F(\Sigma)$ such that distinct formulae
$\varphi(\overline{x})$ are $E$-equivalent to distinct predicates
$R_\varphi(\overline{x})$, then $\nabla_E$ produces a Morleyzation
for $F(\Sigma)$.~$\Box$

\medskip
{\bf Remark 2.2.} Let $F'(\Sigma)$ be the subset of $F(\Sigma)$
consisting of all formulae $\varphi(x_1,$ $\ldots,x_n)$ equivalent
to $\varphi(x_1,\ldots,x_n)\wedge\bigcap\limits_{i<j}\neg
x_i\approx x_j$. As shown in \cite{MT}, each formula $\varphi\in
F(\Sigma)$ is equivalent to a Boolean combination of formulae in
$F'(\Sigma)$. Therefore equivalence relations on $F(\Sigma)$ are
reduced to equivalence relations on $F'(\Sigma)$. Thus, notions
and results below, for $F(\Sigma)$, can be naturally transformed
for $F'(\Sigma)$, and vice versa.

\medskip
We denote by ${\rm SR}(\Sigma)$ the family of all special
relations for $F(\Sigma)$, by ${\rm SR}_f(\Sigma)$ the subfamily
of ${\rm SR}(\Sigma)$ consisting of all special relations $E$ with
finite equivalence $E_{\overline{x}}$-classes only, and by ${\rm
SR}_{\rm ff}(\Sigma)$ the subfamily of ${\rm SR}_f(\Sigma)$
consisting of special relations $E$ with only finitely many,
including $0$, nontrivial $E_{\overline{x}}$-classes, i.e.,
$E_{\overline{x}}$-classes which are not singletons.

\medskip
{\bf Proposition 2.3.} {\em A special relation $E\in{\rm SR}_{\rm
ff}(\Sigma)$ for $F(\Sigma)$ is upward directed if and only if
$E={\rm id}_F(\Sigma)$.}

\medskip
Proof. Clearly, ${\rm id}_F(\Sigma)\in {\rm SR}_{\rm ff}(\Sigma)$
since all equivalence classes for ${\rm id}_F(\Sigma)$ are
singletons.

Conversely, if $E\ne{\rm id}_F(\Sigma)$ then there is
$\overline{x}\in V^\ast$ such that some $E_{\overline{x}}$-class
contains at least two formulae. Assuming that $E$ is upward
directed we obtain, for each $\overline{y}\in V^\ast$,
$E_{\overline{x}\,\hat{\,}\,\overline{y}}$-classes which are not
singletons. Thus, $E\notin{\rm SR}_{\rm ff}(\Sigma)$.~$\Box$

\medskip
{\bf Remark 2.4.} Any special relation $E\in{\rm SR}_{\rm
ff}(\Sigma)$ can be naturally extended till a upward directed
relation $E'\in{\rm SR}_{f}(\Sigma)$ just extending
$E_{\overline{x}\,\hat{\,}\,\overline{y}}$-classes by (finitely
many) formulae in appropriate $E_{\overline{x}}$-classes and
$E_{\overline{y}}$-classes.

\medskip
{\bf Definition.} Let $E$ be a special relation for $F(\Sigma)$.

The relation $E$ is called {\em inessential} if $\nabla_E$
consists of identically true formulae. The relation $E$ is called
{\em essential} if it is not inessential.

For a formula $\varphi=\varphi(\overline{x})\in F(\Sigma)$ the
equivalence class $E_{\overline{x}}(\varphi)$ is called {\em
regular} if the set $\varphi_E$ is consistent.

The relation $E$ is called {\em weakly regular} if for each
formula $\varphi=\varphi(\overline{x})\in F(\Sigma)$ the
equivalence class $E_{\overline{x}}(\varphi)$ is regular.

The relation $E$ is called {\em locally regular} if each finite
part of $\nabla_E$ is consistent.

The relation $E$ is called {\em strongly regular} if $\nabla_E$ is
consistent.

\medskip
The following properties hold:

\medskip
1. (Monotony) If $E,E'\in{\rm SR}(\Sigma)$, $E\subseteq E'$, and
$E'$ is inessential (respectively, weakly regular, strongly
regular) then $E$ inessential (weakly regular, strongly
re\-gu\-lar), too.

\medskip
2. Any inessential relation is strongly regular, but not vice
versa, as Example 2.1 shows.

\medskip
3. Any strongly regular relation is locally regular, and, by
compactness, any locally regular relation is strongly regular.
Besides, any strongly regular relation is weakly regular, but not
vice versa.

\medskip
Indeed, we can consider a naming \begin{equation}\label{eeq1}
\forall \overline{x} (\psi(\overline{x})\leftrightarrow
R(\overline{x}))
\end{equation}
for a consistent formula $\psi(\overline{x})$ generating the set
$\psi_E$, and the formula
\begin{equation}\label{eeq2}
\forall \overline{x} (\neg\psi(\overline{x})\leftrightarrow
(R(\overline{x})\wedge R(\overline{x})))
\end{equation}
producing the set $(\neg\psi)_E$ such that all others
$E_{\overline{y}}$-classes are singletons. Clearly, each set
$\varphi_E$ is consistent. i.e., $E$ is weakly regular. At the
same time, the formulae (\ref{eeq1}) and (\ref{eeq2}) imply
$\forall \overline{x}
(\psi(\overline{x})\leftrightarrow\neg\psi(\overline{x}))$
producing that $\nabla_E$ is inconsistent, i.e., $E$ is not
strongly regular.

\medskip
4. Any singleton $E_{\overline{x}}(\varphi)$ is regular, moreover,
any special relation $E\in{\rm SR}(\Sigma)$ with consistent set
$\bigcup\{\varphi_E\mid |E(\varphi)|>1\}$ is strongly regular. As
a corollary we obtain that any disjoint nonempty sets
$\Phi_i\subset F(\Sigma)$, $i\in I$, of formulae with consistent
set $\{\forall \overline{x}
(\varphi(\overline{x})\leftrightarrow\psi(\overline{x}))\mid
\varphi,\psi\in\Phi_i\mbox{ for some }i\in I\}$ has a strongly
regular relation $E$ such that $\{\Phi_i\mid i\in
I\}\subseteq\{E(\varphi)\mid\varphi\in F(\Sigma)\}$, extending
$\{\Phi_i\mid i\in I\}$ by singletons.

\medskip
5. Any Morleyzation is produced by a strongly regular relation, as
in Example 2.1.

\medskip
By Property 3 we unite the notions of locally regular and strongly
regular special relations obtaining the notion of {\em regular}
special relation. We denote by ${\rm RSR}(\Sigma)$ the subfamily
of ${\rm SR}(\Sigma)$ consisting of all regular special relations.

\medskip
{\bf Proposition 2.5.} {\em For any language $\Sigma$ there is the
largest inessential relation $E_0$ for $F(\Sigma)$. The relation
$E^\ast_0$ equals the relation $\equiv$ for $F(\Sigma)$.}

\medskip
Proof. We denote the relation $\equiv$ by $E^\ast_0$. It naturally
produces a special relation $E_0$ which is inessential by the
definition, since $\varphi\equiv\psi$ if and only if
$\vdash(\varphi\leftrightarrow\psi)$ for any $\varphi,\psi\in
F(\Sigma)$, and thus $\nabla_{E_0}$ consists of identically true
formulae.

Now we note that any proper extension $E\in{\rm SR}(\Sigma)$ of
$E_0$ is essential. Indeed, by the definition there is
$(\varphi,\psi)\in E^\ast$ such that
$\not\vdash(\varphi\leftrightarrow\psi)$. Therefore, by G\"{o}del
completeness theorem, $(\varphi\leftrightarrow\psi)$ is not
identically true. Thus, $E$ is essential.~$\Box$

\medskip
Property 1 and Proposition 2.5 imply:

\medskip
{\bf Corollary 2.6.} {\em A special relation $E\in{\rm
SR}(\Sigma)$ is inessential if and only if $E^\ast\subseteq
E^\ast_0$. In particular, the least special relation ${\rm
id}_{F(\Sigma)}$ is inessential.}

\medskip
{\bf Proposition 2.7.} {\em $(1)$ A relation $E\in{\rm
SR}(\Sigma)$ is inessential if and only if each its subrelation
$E'\in{\rm SR}_{f}(\Sigma)$ is inessential, and if and only if
each its subrelation $E''\in{\rm SR}_{\rm ff}(\Sigma)$ is
inessential.

$(2)$ A relation $E\in{\rm SR}(\Sigma)$ is weakly regular if and
only if each its subrelation $E'\in{\rm SR}_{f}(\Sigma)$ is weakly
regular, and if and only if each its subrelation $E''\in{\rm
SR}_{f}(\Sigma)$ is weakly regular.

$(3)$ A relation $E\in{\rm SR}(\Sigma)$ is regular if and only if
each its subrelation $E'\in{\rm SR}_{f}(\Sigma)$ is regular, and
if and only if each its subrelation $E''\in{\rm SR}_{\rm
ff}(\Sigma)$ is regular.}

\medskip
Proof. (1) follows by Proposition 2.5 and Corollary 2.6.

(2) and (3) hold by compactness since if some $\varphi_E$
(respectively $\nabla_E$) is inconsistent then some finite part of
$\varphi_E$ ($\nabla_E$) is inconsistent producing a special
relation $E''\subseteq E$ in ${\rm SR}_{\rm ff}(\Sigma)$, and in
${\rm SR}_{f}(\Sigma)$ by ${\rm SR}_{\rm ff}(\Sigma)\subset{\rm
SR}_{f}(\Sigma)$, which is not weakly regular {\rm (}regular{\rm
)}.~$\Box$

\section{Elementary special and equivalence relations}

{\bf Definition.} A special relation $E\in{\rm SR}(\Sigma)$ is
called {\em elementary} if there exists a consistent theory $T$ of
given language $\Sigma$ such that
$E=\{(\varphi,\psi,\overline{x})\mid
T\vdash\forall\overline{x}(\varphi(\overline{x})\leftrightarrow\psi(\overline{x}))\}$.

Here the relation $E$ (respectively, $E^\ast$) is called the {\em
special} ({\em equivalence}) {\em relation for} $T$.

The set of all elementary special relations $E\in{\rm SR}(\Sigma)$
is denoted by ${\rm ESR}(\Sigma)$.

\medskip
By the definition any elementary special relation $E\in{\rm
ESR}(\Sigma)$ for $T$ is upward directed, so it is coordinated,
and the relation $E^\ast$ is the equivalence relation $\equiv_T$
modulo $T$, where for any $\overline{x}\in V^\ast$,
$E_{\overline{x}}$ is the restriction of $\equiv_T$ to the set of
formulae $\varphi(\overline{x})$.

Clearly, each relation $E\in{\rm ESR}(\Sigma)$ is regular, and
that relation $E$ is inessential if and only if
$E^\ast=\,\equiv_{T_\Sigma}$ for the least, by inclusion, theory
$T_\Sigma$ of the language $\Sigma$.

Besides, each regular special relation $E$ generates both the
unique consistent theory $T$ with $\nabla_E\vdash T$, which is
axiomatized by $\nabla_E$, and the unique elementary special
relation $E'$ for $T$, containing $E$. Thus the set ${\rm
RSR}(\Sigma)$ of all regular special relations for $F(\Sigma)$ has
a closure operator ${\rm Cl}$ acting by the rule ${\rm Cl}(E)=E'$,
similar to \cite{Tarski}. This closure operator satisfies the
following obvious properties for any $E,E'\in{\rm RSR}(\Sigma)$:

i) $E\subseteq{\rm Cl}(E)$ (Extensiveness),

ii) if $E\subseteq E'$ then ${\rm Cl}(E)\subseteq{\rm Cl}(E')$
(Monotony),

iii) ${\rm Cl}({\rm Cl}(E))={\rm Cl}(E)$ (Idempotency).

The structure $({\rm RSR}(\Sigma),{\rm Cl})$ is extensible till
the structure $({\rm SR}(\Sigma),{\rm Cl}^\ast)$ with the closure
operator ${\rm Cl}^\ast$ producing the special relation $E'={\rm
Cl}^\ast(E)$, for $E\in{\rm SR}(\Sigma)\setminus{\rm
RSR}(\Sigma)$, corresponding the equivalence relation
$(E')^\ast=\,\,\equiv_{T_0}$ for the contradictory theory $T_0$.

\medskip
{\bf Proposition 3.1.} {\em For any relation $E\in{\rm
ESR}(\Sigma)$ there is unique theory $T$ such that
$E^\ast=\,\,\equiv_T$.}

\medskip
Proof. The required theory $T$ exists by the definition of
elementary special relation $E$. Now taking the elementary special
relation $E$ we reconstruct $T$ as the set $T'$ of sentences
$\varphi$ such that $(\varphi,\forall x(x\approx x),\emptyset)\in
E$. Indeed, if $\varphi\in T$ then $\varphi\equiv_T\forall
x(x\approx x)$ producing $\varphi\in T'$. Conversely, if the
sentence $\varphi\in F(\Sigma)$ does not belong to $T$ then
$\varphi$ can not belong to $T'$ since otherwise
$\varphi\equiv_T\forall x(x\approx x)$ implies $\varphi\in
T$.~$\Box$

\medskip
The arguments for Proposition 3.1 show that any relation $E\in{\rm
ESR}(\Sigma)$ uniquely defines a theory $T(E)=T$, and conversely
any consistent theory $T$ uniquely defines an elementary special
relation $E$ with $E^\ast=\,\,\equiv_T$. Moreover, both $T(E)$ and
$\equiv_T$ are uniquely defined by the $E^\ast$-class $E^\ast(1)$
containing identically true formulae. Therefore there are
one-to-one correspondences between the class ${\rm EE}$ of
elementary equivalence relations $E^\ast$, the class $\mathcal{T}$
of consistent theories $T=T(E)$, and the class ${\rm EE}(1)$
consisting of the $E^\ast$-classes $E^\ast(1)$. These
correspondences are denoted by ${\rm EE}\to\mathcal{T}$, ${\rm
EE}\to{\rm EE}(1)$. Additionally we obtain correspondences
$\mathcal{T}\to{\rm EE}$, ${\rm EE}(1)\to{\rm EE}$,
$\mathcal{T}\to{\rm EE}(1)$, ${\rm EE}(1)\to\mathcal{T}$.

Any elementary special relation $E$, for a theory $T=T(E)$,
produces an appro\-pri\-a\-te Lindenbaum--Tarski algebra
\cite{Myers} defined by the partial order $\leq$ in the following
way: for equivalence classes $\equiv_T\!\!(\varphi)$ and
$\equiv_T\!\!(\psi)$, where $\varphi,\psi\in F(\Sigma)$,
$$\equiv_T\!\!(\varphi)\leq\,\,\equiv_T\!\!(\psi)\Leftrightarrow(\varphi\wedge\psi,\varphi)\in E^\ast.$$
We denote this algebra by ${\rm LT}(E)$ and by ${\rm LT}(T)$, and
the class of these algebras by $\mathcal{LT}$.

The algebras in $\mathcal{LT}$ are connected with theories in
$\mathcal{T}$ by the one-to-one co\-r\-re\-s\-pon\-den\-ces
$\mathcal{LT}\to\mathcal{T}$ and $\mathcal{T}\to\mathcal{LT}$ with
the rules ${\rm LT}(T)\mapsto T$ and $T\mapsto {\rm LT}(T)$.

Clearly, homomorphisms between algebras in $\mathcal{LT}$ induce
by these correspondences homomorphisms between elements in
$\mathcal{T}$, in ${\rm EE}$, and in ${\rm EE}(1)$. Therefore we
obtain four categories $\mathcal{LT}$, $\mathcal{T}$,
$\mathcal{EE}$, $\mathcal{EE}(1)$, whose morphisms are
homomorphisms. These categories are connected by functors
$\alpha\to\beta$ above, where $\alpha,\beta\in\{\mathcal{LT}$,
$\mathcal{T}$, $\mathcal{EE}$, $\mathcal{EE}(1)\}$.

Thus we have the following:

\medskip
{\bf Theorem 3.2.} {\em The categories $\mathcal{LT}$,
$\mathcal{T}$, $\mathcal{EE}$, $\mathcal{EE}(1)$ are connected by
the functors $\alpha\to\beta$, where
$\alpha,\beta\in\{\mathcal{LT}$, $\mathcal{T}$, $\mathcal{EE}$,
$\mathcal{EE}(1)\}$.}

\medskip
{\bf Definition} \cite{ZPS}. A theory $T$ is said to be
\emph{$\Delta$-based}\index{Theory!$\Delta$-based}, where $\Delta$
is some set of~formulae without parameters, if any formula of $T$
is equivalent in $T$ to a~Boolean combination of formulae
of~$\Delta$.

For $\Delta$-based theories $T$, it is also said that $T$ has {\em
quantifier elimination}\index{Elimination of quantifiers} or {\em
quantifier reduction}\index{Reduction of quantifiers} modulo
$\Delta$.

Clearly, if $\Delta$ consists of quantifier free formulae then the
$\Delta$-based theory $T$ has quantifier elimination.

\medskip
The following proposition gives an obvious characterization for
the $\Delta$-basedness of a theory.

\medskip
{\bf Proposition 3.3.} {\em Let $\Delta\subseteq F(\Sigma)$. A
theory $T$ is $\Delta$-based if and only if for each formula
$\varphi(\overline{x})\in F(\Sigma)$ there is a
$(\equiv_T)_{\overline{x}}$-class containing both $\varphi$ and a
formula in $\Delta$.}

\medskip
Proposition 3.3 immediately implies:

\medskip
{\bf Corollary 3.4.} {\em A theory $T$ of a language $\Sigma$ has
quantifier elimination if and only if for each formula
$\varphi(\overline{x})\in F(\Sigma)$ there is a
$(\equiv_T)_{\overline{x}}$-class containing both $\varphi$ and a
quantifier free formula.}

\medskip
Clearly, each $E^\ast$-class, for an elementary special relation
$E$, is infinite. Moreover, since there are ${\rm
max}\{\omega,|\Sigma|\}$ identically true sentences $\varphi_1$ of
the language $\Sigma$ and each $E^\ast$-class contains
$\psi\wedge\varphi_1$ together with a formula $\psi$, each
$E^\ast$-class contains ${\rm max}\{\omega,|\Sigma|\}$ formulae.

\medskip
The following natural questions arise:

\medskip
{\bf Question 1.} What can be cardinalities of quotients
$F(\Sigma)/{E^\ast}$ and
$F(\Sigma)_{\overline{x}}/{E_{\overline{x}}}$, for restrictions
$F(\Sigma)_{\overline{x}}$ of $F(\Sigma)$ to the sets of formulae
$\varphi(\overline{x})$, where $\overline{x}\in V^\ast$?

\medskip
{\bf Question 2.} What special relations for $F(\Sigma)$ are
elementary, depending on languages $\Sigma$?

\medskip
{\bf Question 3.} What are hierarchies for elementary special
relations for $F(\Sigma)$, depending on languages $\Sigma$?

\medskip
A partial answer to the Question 3 is represented in the following
section.


\section{Complete and precomplete special and equivalence relations, their completions}

Extending the class $E^\ast(1)$, for an elementary special
relation $E\in{\rm ESR}(\Sigma)$, by maximum many sentences
preserving the consistency we obtain a completion $T'$ of $T=T(E)$
correspondent to an elementary special relation $E'\supseteq E$.
The relation $E'$ for the completion $T'$ is called {\em
complete}, or the {\em completion} of $E$. The set of all complete
relations $E\in{\rm ESR}(\Sigma)$ is denoted by ${\rm
CSR}(\Sigma)$.

\medskip
{\bf Definition.} A relation $E'\in{\rm ESR}(\Sigma)$ is called a
{\em a prime extension} of a relation $E\in{\rm ESR}(\Sigma)$ if
there is a pair $(\varphi,\psi)\notin E$ of formulae such that
$\nabla_E\cup\{\varphi\leftrightarrow\psi\}$ forces $E'$. If $E'$
is the prime extension of $E$ via the pair $(\varphi,\psi)$ we say
that $(\varphi,\psi)$ {\em witnesses} that prime extension, and we
denote the pair $(E,E')$ by $E\prec_{\varphi,\psi} E'$, or simply
$E\prec E'$ if an appropriate pair $(\varphi,\psi)$ is fixed.

A relation $E\in{\rm ESR}(\Sigma)\setminus{\rm CSR}(\Sigma)$ is
called {\em precomplete}, denoted by $E\in{\rm PCSR}(\Sigma)$, if
$E\prec_{\varphi,\psi} E'$ for some pair $(\varphi,\psi)$ and a
complete special relation $E'\in{\rm CSR}(\Sigma)$.

If $E\in{\rm CSR}(\Sigma)$ (respectively, $E\in{\rm
PCSR}(\Sigma)$) then the equivalence relation $E^\ast$ is called
({\em pre}){\em complete}, too.

\medskip
{\bf Example 4.1.} If the language $\Sigma$ consists of unique
unary predicate $R$ then the inessential special relation $E$ is
precomplete obtaining a complete extension $E'\supseteq E$ just by
adding a pair $(\varphi,\forall x\approx x)$ of sentences such
that $\varphi$ asserts a finite cardinality for $R$ and a finite
cardinality for its complement.

Similarly, any extension of a complete theory $T$ by new unary
predicate $R$ produce a precomplete relation $E$ generated by
$\equiv_T$ and tautologies for $\Sigma(T)\cup\{R\}$. The relation
$E$ is extended to a complete one by the pair $(\varphi,\forall
x\approx x)$, where, for instance, $\varphi$ asserts that $R$ is
either empty or has the empty complement.

\medskip
By the definition the number of elementary special relations
extending $E$ is connected with the number of completions and
precomplete extensions of $T(E)$. In particular, by Propositions
1.5--1.7, for certain languages, $E$ can have at most countably
many or at least continuum many completions.

For $E\in{\rm ESR}(\Sigma)$, we denote by ${\rm Comp}(E)$ the set
$\{E'\in{\rm CSR}(\Sigma)\mid E'\supseteq E\}$. Similarly, for a
consistent theory $T$ in the language $\Sigma$ we denote by ${\rm
Comp}(T)$ the set of all complete theories $T'$ in the language
$\Sigma$ which contain $T$.

By the definition the set ${\rm Comp}(T)$ is $d$-definable, and so
closed, in the family $\mathcal{T}_\Sigma$ of all theories of the
language $\Sigma=\Sigma(T)$: ${\rm
Comp}(T)=(\mathcal{T}_\Sigma)_T$.

By Theorem 3.2 there is a natural bijection between ${\rm
Comp}(T(E))$ and ${\rm Comp}(E)$ acting by the {\em canonical} map
$f^{{\rm Comp}(T(E))}\mbox{: } T'\mapsto\,\,\equiv_{T'}$ for
$T'\supseteq T(E)$.

By the definition for any $E\in{\rm ESR}(\Sigma)$, $|{\rm
Comp}(E)|\geq 1$. Moreover, ${\rm Comp}(E)=\{E\}$ and so $|{\rm
Comp}(E)|=1$ for complete $E$. Besides, if $E\in{\rm
ESR}(\Sigma)\setminus{\rm CSR}(\Sigma)$, in particular, if $E$ is
precomplete, then $|{\rm Comp}(E)|\geq 2$.

Indeed, if $E$ is not complete then there is a sentence
$\varphi\in {\rm Sent}(\Sigma)$ such that $T(E)\cup\{\varphi\}$
and $T(E)\cup\{\neg\varphi\}$ are consistent. Therefore $T_1={\rm
Th}(T(E)\cup\{\varphi\})$ and $T_2={\rm
Th}(T(E)\cup\{\neg\varphi\})$ are proper extensions of $T(E)$,
hence special relations $\equiv_{T_1}$ and $\equiv_{T_2}$ are
proper extensions of $E$, implying $|{\rm Comp}(E)|\geq 2$.

Thus we have the following:

\medskip
{\bf Proposition 4.2.} {\em If $E\in{\rm ESR}(\Sigma)$ then
$E\in{\rm CSR}(\Sigma)$ if and only if ${\rm Comp}(E)$ is a
singleton {\rm (}consisting of $\{E\}${\rm )}.}

\medskip
{\bf Proposition 4.3.} {\em For any cardinality $\lambda\geq 2$
there is a special relation $E\in{\rm PCSR}(\Sigma)$ such that
$|{\rm Comp}(E)|=\lambda$.}

\medskip
Proof. It suffices to take a complete theory $T_0$ of a language
$\{c_i\mid i<\lambda\}$ with $\lambda$ distinct constants expanded
by a unary predicate $R$ with the sentence $\exists^{=1}x R(x)$.
Here we assume that $(x\approx
x)\leftrightarrow\bigvee\limits_{i<\lambda} (x\approx c_i)$ if
$\lambda$ is finite. The resulted theory $T$ has a precomplete
special relation $E=\,\,\equiv_T$ with $\lambda$ completions by
sentences $R(c_i)$. If $\lambda$ is infinite there is a completion
by the set of sentences $$\exists
x\left(R(x)\wedge\bigwedge\limits_{i_j\in\{i_1,\ldots,i_k\}\subset\lambda}\neg
(x\approx c_{i_j})\right).$$ In any case we have $|{\rm
Comp}(E)|=\lambda$.~$\Box$

\medskip
The proof above is based on possibilities of interpretation of new
predicate $R$ by definable sets $\varphi(\mathcal{M})$ in initial
models $\mathcal{M}\models T$ via the sentence
$\forall\overline{x}(R(\overline{x})\leftrightarrow\varphi(\overline{x}))$
extending the special relation $E$ by the pair
$(R(\overline{x}),\varphi(\overline{x}))$. In general case,
precomplete special relation becomes complete via an
identification of some two definable sets $\varphi(\mathcal{M})$
and $\psi(\mathcal{M})$.


\medskip
Arguments for the proof of Proposition 4.3 imply the following:

\medskip
{\bf Problem.} {\em Describe possibilities for the cardinalities
of ${\rm Comp}(E)$ depending on the languages $\Sigma$ and
possibilities of identifications.}


\medskip
For a consistent theory $T$ we denote by ${\rm RS}_{\rm Comp}(T)$
and by ${\rm ds}_{\rm Comp}(T)$ the values ${\rm RS}({\rm
Comp}(T))$ and ${\rm ds}({\rm Comp}(T))$, respectively, for the
family ${\rm Comp}(T)$.

These rank and degree values form complexity measured for theories
$T$ via correspondent measures for sets of completions. These
values have upper bounds by the values for $\mathcal{T}_\Sigma$
\cite{MarkhSud}.

\medskip
The following three properties are obvious:

\medskip
1. If $T_1$ and $T_2$ are theories in a language $\Sigma$ with
$T_1\subseteq T_2$ then ${\rm Comp}(T_1)\supseteq{\rm Comp}(T_1)$,
${\rm RS}_{\rm Comp}(T_1)\geq {\rm RS}_{\rm Comp}(T_2)$, and ${\rm
ds}_{\rm Comp}(T_1)\geq {\rm ds}_{\rm Comp}(T_2)$ if $${\rm
RS}_{\rm Comp}(T_1)={\rm RS}_{\rm Comp}(T_2).$$

\medskip
2. If a theory $T$ consists of tautologies in a language $\Sigma$
then ${\rm Comp}(T)=\mathcal{T}_\Sigma$ with the maximal rank
${\rm RS}({\rm Comp}(T))={\rm RS}(\mathcal{T}_\Sigma)$.

\medskip
3. If $T$ is a complete theory then ${\rm Comp}(T)=\{T\}$, ${\rm
RS}({\rm Comp}(T))=0$, $${\rm ds}({\rm Comp}(T))=1.$$

\medskip
4. For any theory $T$ in a language $\Sigma$ and a set $\Phi$ of
$\Sigma$-sentences the set ${\rm Comp}(T)_\Phi$ equals the set
${\rm Comp}(T\cup\Phi)$ of completions for $T\cup\Phi$, and equals
the set ${\rm Comp}(T')$ for the theory $T'={\rm Th}(T\cup\Phi)$
axiomatized by $T\cup\Phi$:
$$
{\rm Comp}(T)_\Phi={\rm Comp}(T\cup\Phi)={\rm Comp}({\rm
Th}(T\cup\Phi)).
$$

%\medskip
Indeed, if $T'\in{\rm Comp}(T)_\Phi$ then $T'$ contains
$T\cup\Phi$ implying $T'\in {\rm Comp}(T\cup\Phi)$ and $T'\in{\rm
Comp}({\rm Th}(T\cup\Phi))$. If $T'\in{\rm Comp}(T\cup\Phi)$ then
$T'$ is a complete theory containing both $T$ and $\Phi$,
therefore $T'\in{\rm Comp}(T)$ with $\Phi\in T'$, i.e., $T'\in{\rm
Comp}(T)_\Phi$. Finally, by the definition of the operators ${\rm
Comp}$ and ${\rm Th}$ we have ${\rm Comp}(T\cup\Phi)={\rm
Comp}({\rm Th}(T\cup\Phi))$.

\medskip
By Property 4 for any set $\Phi$ of $\Sigma$-sentences the
subfamily ${\rm Comp}(T\cup\Phi)$ of ${\rm Comp}(T)$ is
$d$-definable. In particular, if $\Phi$ is finite then ${\rm
Comp}(T\cup\Phi)$ is $s$-definable in ${\rm Comp}(T)$. Therefore,
${\rm Comp}(T\cup\Phi)$ is $E$-closed in ${\rm Comp}(T)$ by
\cite[Theorem 4.3]{dst}.

\medskip
In view of the properties above, Proposition 1.8 and Theorem 1.9
we have the following:

\medskip
{\bf Proposition 4.4.} {\em For any ordinals $\alpha\leq\beta$, if
${\rm RS}({\rm Comp}(T))=\beta$ then $${\rm RS}({\rm Comp}({\rm
Th}(T\cup\{\varphi\})))=\alpha$$ for some {\rm
(}$\alpha$-ranking{\rm )} sentence $\varphi$. Moreover, there are
${\rm ds}({\rm Comp}(T))$ pairwise ${\rm Comp}(T)$-inconsistent
$\beta$-ranking sentences for ${\rm Comp}(T)$, and if
$\alpha<\beta$ then there are infinitely many pairwise ${\rm
Comp}(T)$-inconsistent $\alpha$-ranking sentences for ${\rm
Comp}(T)$.}

\medskip
{\bf Theorem 4.5.} {\em Let \ ${\rm Comp}(T)$ \ be a family of a
countable language \ $\Sigma$ \ and with \ ${\rm RS}({\rm
Comp}(T))=\infty$, $\alpha$ be a countable ordinal,
$n\in\omega\setminus\{0\}$. Then there is a $d_\infty$-definable
subfamily $\mathcal{T}^\ast\subset{\rm Comp}(T)$ such that ${\rm
RS}(\mathcal{T}^\ast)=\alpha$ and ${\rm ds}(\mathcal{T}^\ast)=n$.}

\medskip
The properties above, Proposition 4.1 and Theorem 1.9 can be
naturally trans\-formed for relations $E\in{\rm ESR}(\Sigma)$ and
$E^\ast$ by the canonical map $f^{{\rm Comp}(T(E))}$.

\section{Chains of elementary special and equivalence relations}

Now we consider both chains of elementary special relations $E$
and equivalence relations $E^\ast$, and chains of theories $T(E)$
in a language $\Sigma$.

Clearly, these chains define each others, therefore principally we
deal with chains $C=\{E_i\mid i<\lambda\}$ of strictly increasing
elementary special relations. We will assume that the chains are
{\em prime}, i.e., $E_i\prec E_{i+1}$ for unlimit steps and
$E_i=\bigcup\limits_{j<i}E_j$ for limit steps $i$.

By the definition the cardinalities of these chains have the upper
bounds $|F(\Sigma)|$, i.e., ${\rm max}\{|\Sigma|,\omega\}$. Taking
into consideration these bounds we introduce the following:

\medskip
{\bf Definition.} Let $E,E'\in{\rm ESR}(\Sigma)$ with $E\subset
E'$. If for a cardinality $\lambda$ there is a prime chain
$C=\{E_i\mid i<\lambda+1\}$ such that $E_0=E$ and $E_\lambda=E'$,
we say that the pair $(E,E')$ has a {\em prime $\lambda$-chain}.

The set $\{\lambda\mid(E,E')$ has a prime $\lambda$-chain$\}$ is
called the {\em prime spectrum} for $(E,E')$ and denoted by ${\rm
pSp}(E,E')$.

The cardinality ${\rm pl}(E,E')={\rm sup}({\rm pSp}(E,E'))$ is
called the {\em longness of prime $(E,E')$-chains}.

A relation $E\in{\rm ESR}(\Sigma)$ is called {\em prime
$\lambda$-completeable}, for a cardinality $\lambda$, if there is
a prime $\lambda$-chain $C=\{E_i\mid i<\lambda+1\}$ such that
$E_0=E$ and $E_\lambda\in{\rm CSR}(\Sigma)$, i.e., $\lambda\in
{\rm pSp}(E,E')$ for some $E'\in{\rm CSR}(\Sigma)$.

The cardinality ${\rm cpSp}(E)=\bigcup\limits_{E\subset E'\in{\rm
CSR}(\Sigma)}{\rm pSp}(E,E')$ is called the {\em spectrum of prime
completions} of $E$, and the cardinality $${\rm cpl}(E)={\rm
sup}\{\lambda\mid E\mbox{ is prime
}\lambda\mbox{-completeable}\}$$ is called the {\em longness} of
prime completion of $E$.

We assume that ${\rm cpSp}(E)=\{0\}$ and ${\rm cpl}(E)=0$ for
$E\in{\rm CSR}(\Sigma)$.

\medskip
Clearly, ${\rm pSp}(E,E')$ is nonempty with $0\notin{\rm
pSp}(E,E')$ for any $E\subset E'$, therefore  ${\rm pl}(E,E')$ is
a positive cardinal. Similarly, ${\rm cpSp}(E)$ is nonempty with
$0\notin{\rm cpSp}(E)$ for any $E\in{\rm ESR}(\Sigma)\setminus{\rm
CSR}(\Sigma)$, with ${\rm cpl}(E)>0$. Thus, ${\rm cpl}(E)=0$ if
and only if $E\in{\rm CSR}(\Sigma)$.

By the definition we have $1\in{\rm cpSp}(E)$ if and only if $E$
is precomplete.

\medskip
{\bf Lemma 5.1.} {\em For any ${\rm ESR}(\Sigma)$ the relation
$\prec$ is transitive.}

\medskip
Proof. Let $E_0,E_1,E_2\in{\rm ESR}(\Sigma)$ with
$E_0\prec_{\varphi_1,\psi_1}E_1$,
$E_1\prec_{\varphi_1,\psi_1}E_2$. Since $E_2$ is forced by
$\nabla_{E_0}\cup\{\varphi_1\leftrightarrow\psi_1,\varphi_2\leftrightarrow\psi_2\}$
it suffices to find a pair $(\varphi^\ast,\psi^\ast)\in E^\ast_2$
such that
$\nabla_{E_0}\cup\{\varphi^\ast\leftrightarrow\psi^\ast\}$ forces
both $\varphi_1\leftrightarrow\psi_1$ and
$\varphi_2\leftrightarrow\psi_2$. We can take
$(\varphi_1\leftrightarrow\psi_1)\wedge(\varphi_2\leftrightarrow\psi_2)$
for $\varphi^\ast$ and a tautology for $\psi^\ast$. Clearly,
$\varphi^\ast\leftrightarrow\psi^\ast$ forces the required
formulae.~$\Box$

\medskip
Lemma 5.1 immediately implies:

\medskip
{\bf Corollary 5.2.} {\em For any ${\rm ESR}(\Sigma)$ the relation
$\prec\cup\,\,{\rm id}_{{\rm ESR}(\Sigma)}$ is a partial order.}

\medskip
The following assertion shows that finite prime $(E,E')$-chains of
prime extensions produce initial segments for values ${\rm
pSp}(E,E')$

\medskip
{\bf Proposition 5.3.} {\em If $n\in{\rm pSp}(E,E')$ for a natural
number $n\ne 0$ then $n\setminus\{0\}\subseteq{\rm pSp}(E,E')$.}

\medskip
Proof. Since $\prec$ is transitive by Lemma 5.1, for each finite
prime $(E,E')$-chain $E_0,E_1,E_2,$ $\ldots,E_n$ witnessing
$n\in{\rm pSp}(E,E')$, for $n>1$, the $(E,E')$-chain $E_0,E_2,$
$\ldots,E_n$ is prime, too, implying $(n-1)\in{\rm pSp}(E,E')$.
Continuing the process we obtain $n\setminus\{0\}\subseteq{\rm
pSp}(E,E')$.~$\Box$

\medskip
Proposition 5.3 immediately implies:

\medskip
{\bf Corollary 5.4.} {\em If $E\subset E'$ and ${\rm
pl}(E,E')\in\omega$ then ${\rm pSp}(E,E')=n\setminus\{0\}$ for
some $n\in\omega\setminus\{0,1\}$.}

\medskip
{\bf Corollary 5.5.} {\em If $E\in{\rm ESR}(\Sigma)\setminus{\rm
CSR}(\Sigma)$ and ${\rm cpSp}(E)\cap\omega\ne\emptyset$ then $E$
is precomplete.}

\medskip
{\bf Definition.} Let $E,E'\in{\rm ESR}(\Sigma)$ with $E\subseteq
E'$. We say that $E'$ is {\em finitely axiomatizable} over $E$ if
$E'$ is forced by $\nabla_E\cup\{\varphi\leftrightarrow\psi\}$ for
some $(\varphi,\psi)\in(E')^\ast$, i.e. $E\prec_{\varphi,\psi}
E'$.

\medskip
{\bf Lemma 5.6.} {\em If $E'\in{\rm Comp}(E)$ then $E'$ is
finitely axiomatizable over $E$ if and only if $E'$ is isolated in
${\rm Comp}(E)$ by some formula $\varphi\leftrightarrow\psi$ with
$(\varphi,\psi)\in(E')^\ast$.}

\medskip
Proof. If $E'$ is forced by
$\nabla_E\cup\{\varphi\leftrightarrow\psi\}$ then
$\varphi\leftrightarrow\psi$ forces unique element in ${\rm
Comp}(E)$ implying that $E'$ is isolated. Conversely, if $E'$ is
not isolated in ${\rm Comp}(E)$ then there are no formulae
$\varphi\leftrightarrow\psi$ forcing $E'$ in ${\rm Comp}(E)$.
Therefore $E'$ is not finitely axiomatizable over $E$.~$\Box$

\medskip
{\bf Theorem 5.7.} {\em For any $E\in{\rm ESR}(\Sigma)$ the
following conditions are equivalent:

$1)$ ${\rm cpSp}(E)\subseteq\omega$;

$2)$ ${\rm cpSp}(E)=\{0\}$ or ${\rm cpSp}(E)=n\setminus\{0\}$ for
some $n\in\omega\setminus\{0,1\}$;

$3)$  ${\rm Comp}(E)$ is finite.}

\medskip
Proof. $1)\Rightarrow 3)$. We denote by $Y$ the set of all
$\varphi\in{\rm Sent}(\Sigma)$ such that $T(E)\cup\{\varphi\}$
forces a complete theory $T(E')$, i.e., $E'$ is finitely
axiomatizable over $E$ that is $E'$ is isolated by $\varphi$ in
view of Lemma 5.6. Assume on contrary that ${\rm Comp}(E)$ is
infinite. Then by compactness we have consistent
$Z=T(E)\cup\{\neg\varphi\mid\varphi\in Y\}$. Therefore there is a
completion $T(E'')$ of $Z$. We have $E''\in{\rm Comp}(E)$ which is
not finitely axiomatizable over $E$ that is non-isolated in ${\rm
Comp}(E)$ by Lemma 5.6. It means that each prime $(E,E'')$-chain
is infinite implying ${\rm cpSp}(E)\not\subseteq\omega$.

Having $|{\rm Comp}(E)|<\omega$ we obtain $|{\rm Comp}(E)|=1$, if
$E\in{\rm CSR}(\Sigma)$, and $$|{\rm Comp}(E)|\geq 2,$$ if
$E\notin{\rm CSR}(\Sigma)$, since in the latter case $E$ has at
least two completions.

$3)\Rightarrow 2)$. If ${\rm Comp}(E)$ is finite then completions
of ${\rm Comp}(E)$ are isolated. So they are finitely
axiomatizable. By Proposition 4.2 all prime chains starting with
$E$ are finite and moreover have a finite upper bound for lengths.
If $E\in{\rm CSR}(\Sigma)$ then ${\rm Comp}(E)=\{E\}$ and ${\rm
cpSp}(E)=\{0\}$. Otherwise ${\rm cpSp}(E)=n\setminus\{0\}$ for
some $n\in\omega\setminus\{0,1\}$ in view of Proposition 5.3.

$2)\Rightarrow 1)$ is obvious.~$\Box$

\medskip
{\bf Example 5.8.} We realize the values ${\rm
cpSp}(E)=n\setminus\{0\}$, $n\in\omega$, and the cardinalities for
${\rm Comp}(E)$ by induction in the following way.

We take a $0$-ary relation $P_0$ with values $P_0=0$ and $P_1=1$
producing two completions of the theory $T(E_0)$ for the
inessential special relation $E_0$. Thus, ${\rm cpSp}(E_0)=\{1\}$
and $|{\rm Comp}(E_0)|=2$.

If the elementary special relation $E_n$ is already constructed in
the language $\Sigma_n=\{P_i\mid i\leq n\}$ of $0$-ary predicates
with ${\rm cpSp}(E_n)=\{1,\ldots,n+1\}$ and $|{\rm
Comp}(E_n)|=n+2$, we define the special relation $E_{n+1}$ in the
language $\Sigma_n=\{P_i\mid i\leq n+1\}$ of $0$-ary predicates
extending $T(E_n)$ by the sentences $P_j\to P_{n+1}$, $j<n+1$.

We have ${\rm cpSp}(E_{n+1})=\{1,\ldots,n+2\}$ and $|{\rm
Comp}(E_{n+1})|=n+3$ since there are two possibilities $P_{n+1}=0$
and $P_{n+1}=1$ for $P_0=\ldots=P_n=0$, and unique possibility
$P_{n+1}=1$ if $P_j=1$ for some $j\leq n$. These possibilities
extend both the longest prime chain for ${\rm cpSp}(E_{n})$ by one
step with $P_{n+1}=0$, and ${\rm Comp}(E_{n+1})$ by the value
$P_{n+1}=0$.

Thus the process above produces realizations for ${\rm
cpSp}(E)=n\setminus\{0\}$.

If we continue the process by countably many steps then we obtain
a relation $E\in{\rm ESR}(\Sigma)$ with ${\rm cpSp}(E)=\omega+1$.

Now taking inessential relation $E'\in{\rm ESR}(\Sigma')$ for
$\Sigma'$ consisting of $0$-ary relations $Q_k$, $k\in\omega$, we
obtain $2^\omega$ prime chains for completions of $T(E')$ each of
which has the length omega. Thus, ${\rm cpSp}(E')=\{\omega\}$.

If we combine $E_n$ and $E'$ by the sentences $P_i\to Q_k$, $i\leq
n$, $k\in\omega$, then the resulted relation $E''\in{\rm
ESR}(\Sigma'')$, for $\Sigma''=\{P_i\mid i\leq n\}\cup\{Q_k\mid
k\in\omega\}$, satisfies ${\rm
cpSp}(E')=((n+1)\setminus\{0\})\cup\{\omega\}$.

In view of Theorem 5.7 the possibilities for ${\rm cpSp}(E)$,
where $E\in{\rm ESR}(\Sigma)\setminus{\rm CSR}(\Sigma)$ and
$|\Sigma|\leq\omega$, are exhausted by the values
$n\setminus\{0\}$, $(n\setminus\{0\})\cup\{\omega\}$ for some
$n\in\omega\setminus\{0\}$, $\{\omega\}$, $\omega+1$. Therefore
having realizations above for these possibilities we obtain:

\medskip
{\bf Corollary 5.9.} {\em If $|\Sigma|\leq\omega$ and $E\in{\rm
ESR}(\Sigma)\setminus{\rm CSR}(\Sigma)$ then ${\rm cpSp}(E)$ has
the following possibilities: $n\setminus\{0\}$,
$(n\setminus\{0\})\cup\{\omega\}$ for some
$n\in\omega\setminus\{0\}$, $\{\omega\}$, $\omega+1$.}

\bigskip
\begin{thebibliography}{99}
\bibitem{Peret} M.G.~Peretyatkin, {\it Finitely axiomatizable theories},
Plenum Publ. Corp., New York, 1997.

\bibitem{Er_def} Yu.L.~Ershov, {\it Definability and computability},
Plenum Publ. Corp., New York, 1996.

\bibitem{PiG} A.~Pillay, {\it Geometric stability theory}, Clarendon Press, Oxford,
1996.

\bibitem{RSrank} S.V.~Sudoplatov, {\it Ranks for families of theories
and their spectra}, Lobachevskii Journal of Mathematics, {\bf
42}:12 (2021), 2959--2968.

\bibitem{MarkhSud} N.D.~Markhabatov, S.V.~Sudoplatov, {\it Ranks for families of all theories
of given languages}, Eurasian Mathematical Journal, {\bf 12}:2
(2021), 52--58.

\bibitem{dst} N.D.~Markhabatov, S.V.~Sudoplatov, {\it Definable subfamilies of theories,
related calculi and ranks}, Siberian Electronic Mathematical
Reports, {\bf 17} (2020), 700--714.

\bibitem{MS19} N.D.~Markhabatov, S.V.~Sudoplatov, {\it Algebras for definable families of
theories}, Siberian Electronic Mathematical Reports, {\bf 16}
(2019), 600--608.

\bibitem{M19} N.D.~Markhabatov, {\it Ranks for families of permutation theories}, Bulletin of Irkutsk State
University. Series Mathematics, {\bf 28} (2019), 86--95.

\bibitem{Markh} N.D.~Markhabatov,
{\it Pseudofiniteness of locally free algebras}, Algebra and Model
Theory 12. Collection of papers, Eds. A.~G. Pinus, E.~N.
Poroshenko, S.~V. Sudoplatov, NSTU, Novosibirsk, 2019, 41--46.

\bibitem{SudFP} S.V.~Sudoplatov, {\it Formulas and properties, their links and characteristics},
Mathematics, {\bf 9}:12, 2021, 1391, 16~pp.

\bibitem{KS22} B.Sh.~Kulpeshov, S.V.~Sudoplatov, {\it Properties of ranks for families of strongly minimal
theories},  Siberian Electronic Mathematical Reports, {\bf 19}:1
(2022), to appear.

\bibitem{MT} A.~S. Morozov, D.~A. Tussupov, {\it Minimal predicates for $\Delta$-definability}, Algebra and Logic,
{\bf 59}:4 (2020), 328--340.

\bibitem{MS_top1} N.D.~Markhabatov, S.V.~Sudoplatov, {\it Topologies, ranks, and closures for families of theories.} I,
Algebra and Logic, {\bf 59}:6 (2021), 437--455.

\bibitem{MS_top2} N.D.~Markhabatov, S.V.~Sudoplatov, {\it Topologies, ranks, and closures for families of theories.} II,
Algebra and Logic, {\bf 60}:1 (2021), 38--52.

\bibitem{fpfag} In.I.~Pavlyuk, S.V.~Sudoplatov, {\it Formulas and properties for families of theories of Abelian
groups}, Bulletin of Irkutsk State University. Series Mathematics,
{\bf 36} (2021), 95--109.

\bibitem{ZPS} E.A.~Palyutin, J.~Saffe,
S.S.~Starchenko, {\it Models of superstable Horn theories},
Algebra and Logic, {\bf 24}:3 (1985), 171--210.

\bibitem{Tarski} A.~Tarski, {\it Fundamental Concepts of the Methodology of Deductive Sciences =
Fundamentale Begriffe der Methodologie der deduktiven
Wissenschaften} : [orig. 1930], Logic, Semantics, Metamathematics.
Papers from 1923 to 1938 by Alfred Tarski, transl. by
J.H.~Woodger, Clarendon Press, Oxford, 1956, 60--109.

\bibitem{Myers} D.~Myers, {\it Lindenbaum--Tarski Algebras}, Handbook of Boolean Algebras.
Vol.~3, eds. J.~D.~Monk, R.~Bonnet, North-Holland, Amsterdam, New
York, Oxford, Tokyo, 1989, 1167--1195.



\end{thebibliography}


\end{document}
