\documentclass[11pt,reqno]{amsart}
\usepackage[cp1251]{inputenc}
\usepackage[english,russian]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage[colorlinks]{hyperref}



\usepackage{titlesec}
\titlespacing{\section}{0pt}{2.5ex}{1.5ex}
\titlespacing{\subsection}{0pt}{1.5ex}{1ex}
\titlespacing{\subsubsection}{0pt}{1.5ex}{1ex}
\titleformat{\section}{\large\bfseries\centering}{\thesection}{1em}{}
\titleformat{\subsection}[runin]{\bfseries}{\thesubsection.}{0.5em}{}[.\mbox{\ }]
\titleformat{\subsubsection}[runin]{\bfseries}{\thesubsubsection.}{0.4em}{}[.\mbox{\ }]





%%%%%---------- ORCiD
\newcommand\orcidicon[1]{\href{https://orcid.org/#1}{\includegraphics[scale=0.02]{orcid.pdf}}}


\def\udcs{510.67} %Here the author places classificators of the paper according to Russian classification system
\def\mscs{03C64} %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{example}{Example}
\newtheorem{notat}{Notation}

\hypersetup{linkcolor=blue,filecolor=black,urlcolor=blue, citecolor=blue}

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

\def\semrtop
     {
  \vbox{
     \noindent\logo \hspace{-3mm}  \parbox{12cm}{     \begin{center}
     {\LARGE \textsf{СИБИРСКИЕ ЭЛЕКТРОННЫЕ}} \\[2mm]
     {\LARGE \textsf{МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ}} \\[2mm]
     {\Large \textsf{Siberian Electronic Mathematical Reports}} \\[1mm]
     {\Large\tt{\textcolor{blue}{http://semr.math.nsc.ru}}}\\[1mm]
     \raisebox{1ex}{ISSN 1813-3304 }
     \vspace*{-3mm}
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
%     {\small Sobolev Institute of Mathematics SB RAS}
     \end{center}
}
     \vspace{-1mm}
     \noindent
      \begin{tabular}{c}
     \hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa!} \\
     \hline\hline
     \end{tabular}

     \vspace{3mm}
     {\flushleft\it Vol. 20, No. 2, pp. 144--145 (2023) \hspace{\fill}{\rm\small УДК \udcs \hspace{-4mm}}}
     \newline  %Volume, pages in Russian, filled by Editorial board
     	{\rm\small \textcolor{blue}{https://doi.org/10.33048/semi.2023.20.???}}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaa\;}{\rm\small MSC\ \ \mscs }
  }%\vbox
}




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




\thispagestyle{empty}

\title[Type-preserving formulas in weakly o-minimal theories]{\large Type-preserving formulas in weakly o-minimal theories}


\author[B.SH. KULPESHOV]{{\bf B.Sh. Kulpeshov}\protect\orcidicon{0000-0002-4242-0463}}



\address{Beibut Shaiykovich Kulpeshov  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Institute of Mathematics and Mathematical Modeling
\newline\hphantom{iii} Shevchenko street, 28
\newline\hphantom{iii} 050010, Almaty, Kazakhstan.
}%
\email{\textcolor{blue}{kulpesh@mail.ru}}%
\address{Kazakh British Technical University
\newline\hphantom{iii} Tole bi street, 59
\newline\hphantom{iii} 050000, Almaty, Kazakhstan.}
%
\email{\textcolor{blue}{b.kulpeshov@kbtu.kz}}%


\thanks{\sc Kulpeshov, B.Sh.,
Type-preserving formulas in weakly o-minimal theories}
\thanks{\copyright \ 2023 Kulpeshov B.Sh.}
\thanks{\rm This work was supported by Science Committee of
Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant No.~BR20281002.}
\thanks{\it Received January, 1, 2023, Published December, 31, 2023}%


\semrtop \vspace{1cm}
\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc P.P. Petrov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} In the present paper we study properties of $p$-preserving convex-to-right (left) formulas in weakly
o-minimal theories, where $p$ is a non-algebraic 1-type. It is proved that in the case of an existence of a $p$-preserving convex-to-right (left) formula
that is not equivalence-generating, there exists a $p$-preserving convex-to-left (right) formula
that is also not equivalence-generating; it was shown how it is built from the original formula.\medskip

\noindent{\bf Keywords:} weak o-minimality, type-preserving formula, convex-to-right (left) formula, equivalence relation, convexity rank.
 \end{quote}
}

\bigskip

\section{Preliminaries}
Let $L$ be a countable first-order language. Throughout this paper we consider $L$--structures
and suppose that $L$ contains a binary relation symbol $<$ which is interpreted
as a linear order in these structures. This paper concerns the notion of {\it weak o-minimality}
which was initially deeply studied
by H.D.~Macpherson, D.~Marker and C.~Steinhorn in \cite{mms}.
A subset $A$ of a linearly ordered structure $M$ is {\it convex} if for all $a, b\in A$ and $c\in M$
whenever $a<c<b$ we have $c\in A$.
A {\it weakly o-minimal structure} is a linearly ordered structure
$M=\langle M,=,<,\ldots \rangle$ such that  any definable (with parameters) subset of  $M$
is a union of finitely many convex sets in $M$.
Real closed fields with a proper convex valuation ring provide an important example of weakly o-minimal
 structures.

Let $A$ and $B$ be arbitrary subsets of a linearly ordered structure $M$.
Then the expression $A<B$ means that $a<b$ whenever $a\in A$ and $b\in B$, and $A<b$
means that $A<\{b\}$. For an arbitrary subset $A$ of  $M$ we introduce the following notations:
$A^+:=\{b\in M \mid A<b\}$ and $A^-:=\{b\in M\mid b<A\}$.
For an arbitrary one-type $p$ we denote by $p(M)$ the set of realizations of $p$ in $M$.
If $B\subseteq M$ and $E$ is an equivalence relation on $M$ then we denote by $B/E$
the set of equivalence classes ($E$--classes) which have representatives in $B$.
If $f$ is a function on $M$ then we denote by $Dom(f)$ the domain of $f$.
A theory $T$ is said to be {\it binary} if every formula of the theory $T$ is equivalent in $T$
to a boolean combination of formulas with at most two free variables.

Further throughout the paper we consider an arbitrary complete theory $T$ (if unless otherwise stated),
where $M$ is a sufficiently saturated model of $T$.

\begin{definition}\rm \cite{bbs4} Let $T$ be a weakly o-minimal theory, $M\models T$, $A\subseteq M$,
$p\in S_1(A)$ be non-algebraic.

(1) An $L_A$-formula $F(x, y)$ is said to be {\it $p$-preserving} if there exist  $\alpha$, $\gamma_1$,
$\gamma_2 \in p(M)$ such that
$$[F(M,\alpha)\setminus \{\alpha\}]\cap p(M)\ne\emptyset \mbox{ and } \gamma_1 < F(M, \alpha)\cap p(M) < \gamma_2.$$

(2) A $p$-preserving formula $F(x, y)$ is said to be {\it convex-to-right (left)} if there exists
$\alpha$ $\in$ $p(M)$ such that $F(M,\alpha)\cap p(M)$ is convex, $\alpha$ is the left (right) endpoint of the set
$F(M,\alpha)\cap p(M)$ and $\alpha\in F(M,\alpha)$.
\end{definition}

\begin{definition}\rm \cite{bk}
Let $F(x,y)$ be a $p$--preserving convex-to-right (left) for\-mu\-la.
We say that  $F(x,y)$ is said to be {\it equivalence-generating} if for any  $\alpha, \beta \in
p(M)$ such that $M\models F(\beta, \alpha)$ the following holds: $$\sup [F(M,\alpha)\cap p(M)]=\sup [F(M,\beta)\cap p(M)]$$
$$(\mbox{resp. }\inf [F(M,\alpha)\cap p(M)]=\inf [F(M,\beta)\cap p(M)]).$$
\end{definition}

Let us recall some notions originally introduced in \cite{mms}.
Let $Y\subset M^{n+1}$ be $\emptyset$-definable, let $\pi:
M^{n+1}\to M^n$ be a projection that drops the last
coordinate, and let $Z:=\pi(Y)$. For each $\bar a\in Z$, let
$Y_{\bar a}:=\{ y: (\bar a, y)\in Y \}$. Suppose that for
every $\bar a\in Z$, the set $Y_{\bar a}$ is bounded from above but
has no supremum in $M.$ Let $\sim$
be an $\emptyset$-definable equivalence relation on $M^n$,
defined as follows:
$$\bar a\sim\bar b \mbox{ for all } \bar a, \bar b\in M^n\setminus
Z, \mbox{ and } \bar a\sim \bar b \Leftrightarrow \sup Y_{\bar
a}=\sup Y_{\bar b}, \mbox{ if } \bar a, \bar b\in Z.$$

Let $\overline{Z}:=Z/\sim$, and for each tuple $\bar a\in Z$, we
denote by $[\bar a]$  the $\sim$-class of tuple $\bar a$.
There is a natural $\emptyset$--definable linear ordering
on $M\cup \overline{Z}$, defined as follows.

Let $\bar a\in Z$ and $c\in M$. Then $[\bar a]<c$ if and only if
  $w<c$ for all $w\in Y_{\bar a}$. If $\bar a \not \sim
\bar b$, then there exists some $x\in M$ such that $[\bar
a]<x<[\bar b]$ or $[\bar b]<x<[\bar a]$, and therefore $<$ induces a
linear order on $M\cup \overline{Z}$. We call such a
set $\overline{Z}$ {\it a sort} (in this case,
an $\emptyset$--definable sort) in $\overline{M}$, where
$\overline{M}$ is the Dedekind completion of the structure $M$, and
we view $\overline{Z}$ as naturally embedded in
$\overline{M}$. Similarly, we can obtain a sort in
$\overline{M}$ by considering infima instead of suprema.

Thus, we will consider definable functions from $M$ in
its Dedekind com\-ple\-tion $\overline{M}$, more precisely into
definable sorts of the structure $\overline{M}$, representing
infima or suprema of definable sets.

Let $A,D\subseteq M$, $D$ be infinite,
$Z\subseteq \overline{M}$ be an $A$--definable sort and $f: D\to
Z$ be an  $A$-definable function. We say that $f$ is {\it locally increasing}
({\it  locally decreasing, locally constant}) on $D$ if for any $a\in D$
there exists an infinite interval $J\subseteq D$ containing $\{a\}$ such that $f$ is strictly
increasing (strictly decreasing, constant) on $J$; we also say that $f$ is {\it locally
monotonic} on $D$ if it is locally increasing or locally decreasing on $D$.

Let $f$ be an $A$-definable function on $D\subseteq M$, $E$ be an
$A$-definable equi\-va\-lence relation on $D$. We say that
$f$ is {\it strictly increasing (decreasing) on $D/E$} if for
any $a, b\in D$ with conditions $a<b$ and $\neg E(a, b)$ we have
$f(a)<f(b)$ ($f(a)>f(b)$).



\begin{theorem}\label{f_p} {\rm \cite{jms, smj_2021}}
Let $T$ be a weakly o-minimal theory, $M\models T$, $A\subseteq
M$, $p\in S_1(A)$ be non-algebraic. Suppose that there exists an $A$-definable
function $f$ the domain of which contains the set $p(M)$, and $f$ is not a constant on $p(M)$.
Then $f$ is locally monotonic or
locally constant on $p(M)$ and there exists an $A$-definable equivalence relation $E(x,y)$
partitioning $p(M)$ into infinitely many convex classes such that $f$ is strictly
monotonic on $p(M)/E$.

\end{theorem}

\begin{definition} \label{w}\rm
 \cite{vvv,vvv2} Let $M$ be a weakly
o-mi\-ni\-mal structure, $B, D\subseteq M$, $A\subseteq
\overline{M}$ be a $B$-definable sort and $f: D\to A$
be a $B$-definable function that is locally increasing (decreasing) on $D$.
 We say that $f$ {\it has depth $n$} on $D$ if there are
$B$-definable equivalence relations $E_1(x,y), \ldots, E_n(x,y)$,
partitioning $D$ into infinitely many infinite convex classes,
so that for any $2\le i\le n$ every $E_i$-class is partitioned into
 infinitely many infinite convex $E_{i-1}$-subclasses and
the following holds:


\begin{itemize}
\item $f$ is strictly increasing (decreasing) on each $E_1$-class,

\item $f$ is locally decreasing (increasing) on $D/E_k$ for
any odd $k\le n$ (or the same, $f$ is strictly decreasing (increasing) on every $E_{k+1}(a,M)/E_k$ for any $a\in D$),
\item $f$ is locally increasing (decreasing) on $D/E_k$ for
any even $k\le n$,
\item $f$ is strictly monotonic on $D/E_n$.

\end{itemize}
In this case, the function $f$ is called {\it locally
 increasing (decreasing) of depth $n$}.
\end{definition}

Obviously, a strictly increasing (decreasing) function is {\it locally increasing (decreasing) of depth $0$}.

\begin{theorem}\label{dep_th} {\rm \cite{vvv2}}  Let $T$ be a weakly o-mi\-ni\-mal theory.
Then any definable function into a definable sort has finite depth.
\end{theorem}

In \cite{k_2015},  Definition  \ref{w} was extended by introducing the notion of {\it a locally constant
function of depth $n$}, i.e., in  Definition  \ref{w} the function $f$ is a constant on every
$E_1$-class. Note that in this case the function $f$ can be either locally increasing
or locally decreasing on $D/E_1$.

\begin{definition}\rm \cite{k1}
Let $T$ be a weakly o-minimal theory, $M\models T$, $A\subseteq M$.
{\it The rank of convexity of the set}
$A$ $(RC(A))$ is defined as follows:

1) $RC(A)=-1$ if $A=\emptyset$.

2) $RC(A)=0$ if $A$ is finite and non-empty.

3) $RC(A) \geq 1$ if $A$ is infinite.

4) $RC(A) \geq  \alpha  +  1$ if there exist a parametrically
definable equivalence relation $E(x,y)$ and an infinite sequence of elements
$b_i\in A, i\in \omega$, such that:
\begin{itemize}
\item For every  $i, j\in \omega$ whenever $i\ne j$ we have  $M \models \neg E(b_i, b_j)$;
\item For every $i\in \omega$, $RC(E(x,  b_i))
\geq \alpha$ and $E(M,b_i)$ is a convex subset of $A$.
\end{itemize}

5) $RC(A) \geq \delta$ if $RC(A)
\geq \alpha$ for all $\alpha < \delta$, where $\delta$ is a limit ordinal.

If $RC(A) = \alpha$ for some $\alpha$, we say that $RC(A)$ is defined. Otherwise (i.e. if $RC(A))$ $\geq$
$\alpha$ for all $\alpha$), we put $RC(A) =\infty$.

{\it The rank of convexity of a formula} $\phi(x,\bar a)$, where $\bar a\in M$, is defined as the rank
of convexity of the set $\phi(M,\bar a)$, i.e. $RC(\phi(x,\bar a)):=RC(\phi(M,\bar a))$.

{\it The rank of convexity of an 1-type} $p$ is defined as the rank of convexity of the set $p(M)$, i.e. $RC(p):=RC(p(M))$.
\end{definition}

In particular, a theory has convexity rank $1$ if there are no
definable (with parameters) equivalence relations with infinitely
many infinite convex classes. Clear\-ly, each o-minimal theory has
convexity rank $1$.
\vskip 3mm

In \cite{bk} properties of $p$-preserving convex-to-right (left) formulas were stu\-died.
In particular, it was established that some of these formulas generate an equivalence relation
with infinite convex classes. These results were used at studying the number of pairwise
non-isomorphic countable models for a weakly o-minimal theory in series of papers \cite{KS_2017}--\cite{Izv_2020}.
Here we continue studying properties of $p$-preserving convex-to-right (left) formulas.
In par\-ti\-cu\-lar, we prove the following: if there exists a $p$-preserving convex-to-right (left) formula that is
not equivalence-generating then there exists a $p$-preserving convex-to-left (right) formula that
is also not equivalence-ge\-ne\-ra\-ting (Theorem \ref{th_neg}).
We show how such a formula is constructed from an existing one.



\section{Properties of $p$-preserving formulas}

\begin{fact} Let $T$ be a weakly o-minimal theory, $M\models T$, $A\subseteq M$,
$p\in S_1(A)$ be non-algebraic. Then either $p(M')$ is densely ordered or $p(M')$
is discretely ordered for any $M'\succeq M$.
\end{fact}

\begin{example}\rm Let $M:=\langle \mathbb{Q}, <, f^1\rangle$
be a linearly ordered structure, where  $\mathbb{Q}$ is the set of rational numbers,
and $f(x)=x+1$. Obviously, $f$ is strictly increasing on $M$. It can be proved that $M$ is an o-minimal 1-transitive structure. Let
$p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$, $p$  is non-algebraic and $p(M)$ is densely ordered.
Consider the following formula: $$F(x,y):=y\le x\le f(y).$$

Obviously, $F(x,y)$ is a $p$-preserving convex-to-right formula that is not equivalence-generating.

Let $G(x,y):=F(y,x)$. Since $y\le f(x)$ iff $f^{-1}(y)\le x$,
$$G(x,y)\equiv f^{-1}(y)\le x\le y$$ and $G(x,y)$ is a $p$-preserving
convex-to-left formula that is also not equi\-va\-lence-generating.

If $M=\langle \mathbb{Z},<f^1\rangle$, where $\mathbb{Z}$ is the set of integers, then
$M$ also is an o-minimal 1-transitive structure. Obviously, $p(x):=\{x=x\}\in S_1(\emptyset)$ and
$p(M)$ is discretely ordered. Consider the following formulas:
$$F_1(x,y):=y\le x< f(y), \qquad F_2(x,y):=y\le x< f^2(y).$$

Since $F_1(M,\alpha)\setminus\{\alpha\}=\emptyset$, $F_1(x,y)$ is not $p$-preserving.
But $F_2(x,y)$ is a $p$-preserving convex-to-right formula.
\end{example}

\begin{proposition}\label{prop_w1} Let $T$ be a weakly o-minimal theory of convexity rank 1, $M\models T$,
$A\subseteq M$, $p\in S_1(A)$  non-algebraic,   $F(x,y)$ a $p$-pre\-ser\-ving con\-vex-to-right
(left) formula. Then for any $\alpha\in p(M)$ the following holds:
$$M\models \forall x [F(x,\alpha)\land x\ne \alpha\to \exists z (F(z,x)\land\neg F(z,\alpha))].$$
\end{proposition}

\begin{proof} Without loss of generality, suppose that
$F(x,y)$ is convex-to-right. Since $T$ has convexity rank 1 and $p(M)$ is 1-indiscernible over $A$,
the function $f(y):=\sup F(M,y)$ is strictly monotonic on $p(M)$. Prove that $f$ is strictly
increasing on $p(M)$. Assume the contrary: $f$ is strictly decreasing on $p(M)$.
Then consider the following formula: $$\phi(x):=\forall y [y<x\to F(x,y)].$$

Obviously, we have $\phi(M)\cap p(M)\ne\emptyset$ and $\neg\phi(M)\cap p(M)\ne\emptyset$,
whence $p(M)$ is not 1-indiscernible over $A$. Consequently, $f$ is strictly increasing on $p(M)$.

Take an arbitrary $\alpha\in p(M)$ and consider $F(M,\alpha)$.
Take an arbitrary $\beta\in F(M,\alpha)$ such that $\beta\ne\alpha$. Obviously, $\alpha<\beta$.
Since $f$ is strictly increasing on $p(M)$, $f(\alpha)<f(\beta)$, whence there exists
$\gamma\in F(M,\beta)\setminus F(M,\alpha)$. \end{proof}


\begin{proposition}\label{prop_w2} Let $T$ be a weakly o-minimal theory of convexity rank 1, $M\models T$,
$A\subseteq M$, $p\in S_1(A)$ be non-algebraic, $F(x,y)$ be a $p$-preserving convex-to-right
(left) formula. Then $G(x,y):=F(y,x)$ is a $p$-preserving convex-to-left (right) formula.
\end{proposition}

\begin{proof} Without loss of generality, suppose that
$F(x,y)$ is convex-to-right. Take an arbitrary $\alpha\in p(M)$.
Since $f(y):=\sup F(M,y)$ is strictly increasing on $p(M)$,
$F(\alpha, M)$ is convex, $\alpha\in F(\alpha, M)$
and there exists $\gamma_1\in p(M)$ such that $\gamma_1<F(\alpha, M)$. And also for any $\beta\in p(M)$
with $\beta>\alpha$ we have $f(\beta)>\alpha$, i.e. $\alpha\not\in F(M,\beta)$,
whence $\alpha$ is the right endpoint of $F(\alpha,M)$.
Thus, $G(x,y)$ is $p$-preserving convex-to-left. \end{proof}



\begin{example}\label{ex_wom2} \rm Let $M:=\langle M, <, E^2, f^1\rangle$
be a linearly ordered structure, where  $M=\mathbb{Q}\times\mathbb{Q}$ is ordered
lexicographically.

We define $E$ as follows: for any $a=(a_1,a_2), b=(b_1,b_2)\in M$ we have $E(a,b)$
iff $a_1=b_1$. Obviously, $E(x,y)$ is an equivalence relation partitioning $M$ into
infinitely many infinite convex classes so that the induced ordering on $E$-classes is dense.

We define $f$ as follows: for any $a=(a_1,a_2)\in M$ we have $$f(a)=(a_1+1, -a_2).$$ 

Then $f$
is strictly decreasing on each $E$-class and $f$ is strictly increasing on $M/E$.

It can be proved that $T=Th(M)$ is a weakly o-minimal theory, and $M$ is 1-transitive.  

Let $p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$, $p$ is non-algebraic and $p(M)=M$.
Consider the following for\-mu\-la: $$F(x,y):=y\le x\le f(y).$$

Obviously, $F(x,y)$ is a $p$-preserving convex-to-right formula
that is not equivalence-ge\-ne\-ra\-ting.
Observe that for any $\alpha\in p(M)$ there exists $\beta\in p(M)$ such that $\alpha<\beta$
and $$M\models F(\beta, \alpha)\land \exists x [F(x,\alpha)\land \neg F(x, \beta)].$$

Let $G(x,y):=F(y,x)$. Since $f$ is strictly increasing on $M/E$, $G(x,y)$ is $p$-preserving.
Observe also that $F(\gamma, M)$ consists of two convex sets for any $\gamma\in p(M)$.
Then $G(M,\gamma)$ is not convex, whence  $G(x,y)$ is not  convex-to-left. Let
$$G'(x,y):=\exists t [F(y,t)\land t\le x\le y].$$
It can be checked that $G'(x,y)$ is a $p$-preserving convex-to-left formula that is not equivalence-generating.

Observe that $F(M,\alpha)$ has the right endpoint $f(\alpha)$ in $M$, but $G'(M,\alpha)$ has no a left endpoint in $M$.

Consider the following formulas:
$$F_1(x,y):=\exists z [F(z,y)\land F(x,z)],$$
$$F_n(x,y):=\exists z [F_{n-1}(z,y)\land F(x,z)], n\ge 2,$$
$$G'_1(x,y):=\exists z [G'(z,y)\land G'(x,z)],$$
$$G'_n(x,y):=\exists z [G'_{n-1}(z,y)\land G'(x,z)], n\ge 2.$$

Obviously, for any $a\in p(M)$ we have the following:
$$F(M,a)\subset F_1(M,a)\subset \ldots \subset F_n(M,a)\subset \ldots \mbox{ and}$$
$$G'(M,a)\subset G'_1(M,a)\subset \ldots \subset G'_n(M,a)\subset \ldots.$$

Consider the following set: $$V_{p,F}(a):=\{b\in p(M)\mid b\in F_n(M,a)\cup G_n(M,a) \mbox{ for some } n\ge 1\}.$$
We call this set by {\it $F$-component of an element $a$ in $p$}.
Further we consider the following set of formulas:
$$S(x,a):=p(x)\cup \{x>a\}\cup \{\neg F_i(x,a)\mid i\in\omega\setminus\{0\}\}.$$  

It is locally consistent, and can be extended to a non-isolated type $p'\in S_1(\{a\})$.
Then there exists $M'\succ M$ in which $p'$ is realized. Take an arbitrary $b\in p'(M')$.
Then $V_{p,F}(a)<b$. Moreover, $V_{p,F}(a)<V_{p,F}(b)$. Similarly, we can realize the type extending $S(x,b)$.
Thus, there exists a model of  $T$ in which there are arbitrarily many such $F$-components,
these components can be both discretely and densely ordered between themselves. Whence we obtain that 
 $T$ has $2^{\omega}$ countable models. 
\end{example}

\begin{example}\label{ex_wom3} \rm Let $M:=\langle M, <, E_1^2, E_2^2, f^1\rangle$
be a linearly ordered structure, where  $M=\mathbb{Q}^3$ is ordered
lexicographically.

We define $E_1$ and $E_2$ as follows: for any $a=(a_1,a_2,a_3), b=(b_1,b_2,b_3)\in M$ we have $E_1(a,b)$
iff $a_1=b_1\land a_2=b_2$, and $E_2(a,b)$
iff $a_1=b_1$. Obviously, $E_1(x,y)$ and $E_2(x,y)$  are equivalence relations partitioning $M$ into
infinitely many infinite convex classes so that $E_1(M,a)\subset E_2(M,a)$ for any $a\in M$.

We define $f$ as follows: for any $a=(a_1,a_2,a_3)\in M$ we have $$f(a)=(a_1+1, -a_2,a_3).$$
 Then $f$
is strictly increasing on each $E_1$-class, $f$ is strictly decreasing on $E_2(M,a)/E_1$ for each $a\in M$ and $f$ is strictly increasing on $M/E_2$.

It can be proved that $Th(M)$ is a weakly o-minimal theory.
Let $p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$ and $p$ is non-algebraic.
Consider the following for\-mu\-la: $$F(x,y):=y\le x\le f(y).$$

Obviously, $F(x,y)$ is a $p$-preserving convex-to-right formula
that is not equivalence-ge\-ne\-ra\-ting.
Here $F(\gamma, M)$ consists of three convex sets for any $\gamma\in p(M)$.
\end{example}

\begin{example}\label{ex_womn}\rm Let $M:=\langle M, <, E_1^2, E_2^2,\ldots, E_{n-1}^2, f^1\rangle$
be a linearly ordered structure, where  $M=\mathbb{Q}^n$ is ordered
lexicographically.

We define $E_k$ for each $1\le k\le n-1$ as follows: for any
$a=(a_1,a_2,\ldots, a_n)$, $b=(b_1,b_2,\ldots, b_n)\in M$ we have $$E_k(a,b)
\mbox{ iff } a_1=b_1\land a_2=b_2\land \ldots \land a_{n-k}=b_{n-k}.$$

Obviously, $E_k(x,y)$
is an equivalence relation partitioning $M$ into
infinitely many infinite convex classes for each $1\le k\le n-1$ so that
$$E_1(M,a)\subset E_2(M,a)\subset \ldots \subset E_{n-1}(M,a)$$ for any $a\in M$.

We define $f$ as follows: for any $a=(a_1,a_2,\ldots, a_n)\in M$ we have
$$f(a)=(a_1+1, -a_2,a_3, \ldots, (-1)^{i+1}a_i, \ldots, (-1)^{n+1} a_n).$$
Then $f$ is strictly increasing on $M/E_{n-1}$. If $n$ is even then $f$ is strictly
decreasing on each $E_1$-class, and $f$ is strictly increasing (decreasing) on $E_{k+1}(M,a)/E_k$
for each $a\in M$, where $1\le k\le n-2$ and $k$ is odd (even).  If $n$ is odd then $f$ is strictly
increasing on each $E_1$-class, $f$ is strictly decreasing (increasing) on $E_{k+1}(M,a)/E_k$
for each $a\in M$, where $1\le k\le n-2$ and $k$ is odd (even).

It can be proved that $Th(M)$ is a weakly o-minimal theory.
Let $p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$ and $p$ is non-algebraic.
Consider the following for\-mu\-la: $$F(x,y):=y\le x\le f(y).$$

Obviously, $F(x,y)$ is a $p$-preserving convex-to-right formula
that is not equivalence-ge\-ne\-ra\-ting.
Here $F(\gamma, M)$ consists of $n$ convex sets for any $\gamma\in p(M)$.
\end{example}

Thus, we established the following:
\begin{proposition} For each natural $n\ge 1$ there exist a weakly o-minimal theory $T$, $M\models T$,
non-algebraic $p\in S_1(\emptyset)$ and a $p$-preserving convex-to-right (left) formula $F(x,y)$ such that
for any $\gamma\in p(M)$ the set $F(\gamma, M)$ consists of $n$ convex sets in $p(M)$.
\end{proposition}



\begin{notat}\rm Let $E(x,y)$ be an equivalence relation with convex classes on $M$, and
$f: M\to \overline{M}$ be a function to a definable sort. Take arbitrary $a,b\in M$ and
suppose that $f(b)\in \overline{M}\setminus M$.
We write $E^*(a, f(b))$ if $a<f(b)$ and
there exists $c\in M$ such that $f(b)<c$ and $E(a,c)$.
We also write $E^*(f(b),a)$ if $f(b)<a$ and there exists $c\in M$ such that $c<f(b)$ and $E(c,a)$.
\end{notat}

\begin{proposition}\label{prop_locmon}
 Let $T$ be a weakly o-minimal theory, $M\models T$,
$A\subseteq M$, $p\in S_1(A)$ non-algebraic, $F(x,y)$ a $p$-preserving convex-to-right
(left) formula. Suppose that the function $f(y):=\sup F(M,y)$ is locally monotonic of depth $n$ on $p(M)$
for some $n\in\omega$. Then

(1) $n$ is even (odd) iff $f$ is locally increasing (decreasing) on $p(M)$;

(2) $F(\gamma, M)$ consists  of $n+1$ convex sets in $p(M)$.
\end{proposition}

\begin{proof} By the hypothesis $f(y):=\sup F(M,y)$ is locally
monotonic of depth $n$ on $p(M)$ for some $n\in\omega$. Consequently, there exist
$A$-definable equi\-va\-lence relations $E_1(x,y), \ldots, E_n(x,y)$ partitioning $p(M)$ into
infinitely many infinite convex classes so that $E_1(M,a)\subset \ldots \subset E_n(M,a)$
for any $a\in p(M)$.

Prove (1). If $n$ is even then we assert that $f$ is locally increasing on $p(M)$. Assume the contrary:
$f$ is locally decreasing on $p(M)$. Then $f$ is strictly decreasing on each $E_1$-class,
$f$ is locally increasing on $p(M)/E_k$ for every odd $k\le n$, $f$ is locally decreasing
on $p(M)/E_k$ for every even $k\le n$. Whence we obtain that $f$ is strictly decreasing on $p(M)/E_n$.
Then we have a contradiction with the 1-indiscernibility of $p(M)$.

Similarly, we can establish that if $n$ is odd then $f$ is locally  decreasing on $p(M)$.

Let's prove (2). If $f$ is locally monotonic of depth 1 then by (1) $f$ is locally decreasing on $p(M)$,
i.e. $f$ is strictly decreasing on each $E_1$-class and $f$ is strictly increasing on $p(M)/E_1$.
Take an arbitrary $\gamma\in p(M)$ and consider $E_1(M,\gamma)$. Then
there exist $a_1, a_2\in p(M)$ such that $$a_1<a_2, E_1(a_1, a_2), f(a_2)<\gamma<f(a_1),
E_1^*(\gamma, f(a_1)) \mbox{ and } E_1^*(f(a_2), \gamma).$$

Then we assert that the following convex
sets are contained in $F(\gamma, M)$: the set definable by $\neg E_1(x,a_2)\land a_2<x\le \gamma$
and a nonempty proper convex subset of the set definable by $E_1(x,a_2)\land x<a_2$. These convex sets
are separable by the formula $E_1(x,a_2)\land x>a_2$. Thus, $F(\gamma, M)$ consists of two convex sets in $p(M)$.

{\it Case A.} $n$ is even.

Then by (1) $f$ is
locally increasing on $p(M)$. Take an arbitrary $\gamma\in p(M)$ and consider $E_1(M,\gamma)$.
Then there exist $a_1, a_2\in p(M)$ such that $$a_1<a_2, E_1(a_1, a_2), f(a_1)<\gamma<f(a_2),
E_1^*(f(a_1), \gamma) \mbox{ and } E_1^*(\gamma, f(a_2)).$$

Then we assert that the following convex sets are contained in $F(\gamma, M)$:
the sets definable by $\neg E_n(x,a_2)\land a_2<x\le \gamma$, $E_k(x,a_2)\land \neg E_{k-1}(x,a_2)\land x<a_2$
for every even $1< k\le n$,
$E_k(x,a_2)\land \neg E_{k-1}(x,a_2)\land x>a_2$ for every odd $2< k\le n$, and
a nonempty proper convex subset of the set definable by $E_1(x,a_2)\land x>a_2$.
Observe that these convex sets are definably separable. For example,
$\neg E_n(x,a_2)\land a_2<x\le \gamma$ and $E_n(x,a_2)\land \neg E_{n-1}(x,a_2)\land x<a_2$
are separable by the formula $$E_n(x,a_2)\land \neg E_{n-1}(x,a_2)\land x\ge a_2.$$

Thus, $F(\gamma, M)$ consists of $n+1$ convex sets in $p(M)$.

{\it Case B.} $n$ is odd. This case considered similarly. \end{proof}

\begin{example}\label{ex_wom22} \rm Let $M:=\langle M, <, E^2, R^2\rangle$
be a linearly ordered structure, where  $M=\mathbb{Q}\times\mathbb{Q}$ is ordered
lexicographically.

We define $E$ as follows: for any $a=(a_1,a_2), b=(b_1,b_2)\in M$ we have $E(a,b)$
iff $a_1=b_1$. Obviously, $E(x,y)$ is an equivalence relation partitioning $M$ into
infinitely many infinite convex classes so that the induced ordering on $E$-classes is dense.

We define $R$ as follows: for any $a=(a_1,a_2), b=(b_1,b_2)\in M$ we have $$R(b,a) \mbox{ iff } a_1\le b_1<a_1+\sqrt{2}.$$

Then $r(y):=\sup R(M,y)$ is locally constant on $M$, and more exactly, $r$ is constant on each $E$-class and
$r$ is strictly increasing on  $M/E$ (i.e. $r$ is locally constant of depth 1 on $M$).

It can be proved that $Th(M)$ is a weakly o-minimal theory.
Let $p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$, $p$ is non-algebraic and $p(M)=M$.
Also, $R(x,y)$ is a $p$-preserving convex-to-right formula
that is not equivalence-ge\-ne\-ra\-ting. Observe that $R(\gamma,M)$ is convex for
any $\gamma\in p(M)$.
\end{example}

\begin{example}\label{ex_wom32} \rm Let $M:=\langle M, <, E_1^2, E_2^2, R^2\rangle$
be a linearly ordered structure, where  $M=\mathbb{Q}^3$ is ordered
lexicographically.

We define $E_1$ and $E_2$ as follows: for any $a=(a_1,a_2,a_3), b=(b_1,b_2,b_3)\in M$ we have $E_1(a,b)$
iff $a_1=b_1\land a_2=b_2$, and $E_2(a,b)$
iff $a_1=b_1$. Obviously, $E_1(x,y)$ and $E_2(x,y)$  are equivalence relations partitioning $M$ into
infinitely many infinite convex classes so that $E_1(M,a)\subset E_2(M,a)$ for any $a\in M$.

We define $R$ as follows: for any $a=(a_1,a_2,a_3), b=(b_1,b_2,b_3)\in M$ we have
$$R(b,a) \mbox{ iff } b_1=a_1+1 \mbox{ and } b_2<-a_2+\sqrt{2}.$$

Then $r(y):=\sup R(M,y)$ is locally constant on $M$, and more exactly, $r$ is constant on each $E_1$-class,
$r$ is strictly decreasing on $E_2(M,a)/E_1$ for each $a\in M$ and $r$ is strictly increasing on  $M/E_2$
(i.e. $r$ is locally constant of depth 2 on $M$).

It can be proved that $Th(M)$ is a weakly o-minimal theory.
Let $p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$, $p$ is non-algebraic and $p(M)=M$.
Also, $R(x,y)$ is a $p$-preserving convex-to-right formula
that is not equivalence-ge\-ne\-ra\-ting.
Here $R(\gamma,M)$ consists of two convex sets for any $\gamma\in p(M)$.
\end{example}

\begin{example}\label{ex_womn2}\rm Let $M:=\langle M, <, E_1^2, E_2^2,\ldots, E_n^2, f^1\rangle$
be a linearly ordered structure, where  $M=\mathbb{Q}^{n+1}$ is ordered
lexicographically.

We define $E_k$ for each $1\le k\le n$ as follows: for any
$a=(a_1,a_2,\ldots, a_n$, $a_{n+1})$, $b=(b_1,b_2,\ldots, b_n, b_{n+1})\in M$ we have $$E_k(a,b)
\mbox{ iff } a_1=b_1\land a_2=b_2\land \ldots \land a_{n-k+1}=b_{n-k+1}.$$

Obviously, $E_k(x,y)$
is an equivalence relation partitioning $M$ into
infinitely many infinite convex classes for each $1\le k\le n$ so that
$E_1(M,a)\subset E_2(M,a)\subset \ldots \subset E_n(M,a)$ for any $a\in M$.

We define $R$ as follows: for any $a=(a_1,\ldots, a_n,a_{n+1})$, $b=(b_1,\ldots, b_n$, $b_{n+1})\in M$
we have $$R(b,a) \mbox{ iff } b_1=a_1+1, b_2=-a_2, \ldots, b_i=(-1)^{i+1}a_i, \ldots, b_{n-1}=(-1)^na_{n-1},$$
and $b_n<(-1)^{n+1}a_n+\sqrt{2}$.
Then $r(y):=\sup R(M,y)$ is locally constant on $M$, $r$ is constant on each $E_1$-class
and $r$ is strictly increasing on $M/E_n$. If $n$ is even then $r$ is  strictly
decreasing (increasing) on $E_{k+1}(M,a)/E_k$
for each $a\in M$, where $1\le k\le n-1$ and $k$ is odd (even).  If $n$ is odd then
$r$ is strictly increasing (decreasing) on $E_{k+1}(M,a)/E_k$
for each $a\in M$, where $1\le k\le n-1$ and $k$ is odd (even).
Here $r$ is locally constant of depth $n$ on $M$.

It can be proved that $Th(M)$ is a weakly o-minimal theory.
Let $p(x):=\{x=x\}$. Obviously, $p\in S_1(\emptyset)$, $p$ is non-algebraic and $p(M)=M$.
Also, $R(x,y)$ is a $p$-preserving convex-to-right formula
that is not equivalence-ge\-ne\-ra\-ting.
Here $R(\gamma,M)$ consists of $n$ convex sets for any $\gamma\in p(M)$.
\end{example}

\begin{proposition}\label{prop_locon}
 Let $T$ be a weakly o-minimal theory, $M\models T$,
$A\subseteq M$, $p\in S_1(A)$ non-algebraic, $F(x,y)$ a $p$-preserving convex-to-right
(left) formula. Suppose that the function $f(y):=\sup F(M,y)$ is locally constant of depth $n$ on $p(M)$
for some natural $n\ge 1$. Then

(1) there exist $A$-definable equivalence relations $E_1(x,y), \ldots, E_n(x,y)$
par\-ti\-tio\-ning $p(M)$ into infinitely many infinite convex classes so that
$E_1(M,a)\subset \ldots E_n(M,a)$ for any $a\in p(M)$, $f$ is strictly increasing on $p(M)/E_n$
and $n$ is even (odd) iff $f$ is locally decreasing (increasing) on $p(M)/E_k$ for every odd $k\le n$
and $f$ is locally increasing (decreasing) on $p(M)/E_k$ for every even $k\le n$;

(2) $F(\gamma, M)$ consists  of $n$ convex sets in $p(M)$.
\end{proposition}

\begin{proof} Prove (1).
By the hypothesis $f(y):=\sup F(M,y)$ is locally
constant of depth $n$ on $p(M)$ for some natural $n\ge 1$. Consequently, there exist
$A$-definable equivalence relations $E_1(x,y), \ldots, E_n(x,y)$ partitioning $p(M)$ into
infinitely many infinite convex classes so that $E_1(M,a)\subset \ldots \subset E_n(M,a)$
for any $a\in p(M)$, and $f$ is constant on each $E_1$-class.

Without loss of generality suppose $n$ is even. Since $f$ has depth $n$,
$f$ is locally increasing or locally decreasing on $p(M)/E_k$ for every odd $k\le n$
and $f$ is locally decreasing or locally increasing on $p(M)/E_k$ for every even $k\le n$.
If $f$ is locally increasing on $p(M)/E_1$ then $f$ is strictly decreasing on $p(M)/E_n$,
whence we have a contradiction with the 1-indiscernibility of $p(M)$. Consequently,
$f$ is locally decreasing on $p(M)/E_k$ on every odd $k\le n$ and $f$ is locally
increasing on $p(M)/E_k$ on every even $k\le n$. Similar reasons for case when $n$ is odd.

Let's prove (2). If $f$ is locally constant of depth 1 then by (1) $f$ is constant on each $E_1$-class
and $f$ is strictly increasing on $p(M)/E_1$.
Take an arbitrary $\gamma\in p(M)$. Then
there exist $a_1, a_2\in p(M)$ such that $a_1<a_2$, $\neg E_1(a_1, a_2)$ and $f(a_1)<\gamma<f(a_2)$.
Since $f$ is strictly increasing on $p(M)/E_1$, there exist a nonempty proper convex subset of the set
definable by the formula $\neg E_1(x,a_1)\land x>a_1$ that defines $F(\gamma, M)$.
  Thus, $F(\gamma, M)$ is convex in $p(M)$.

{\it Case A.} $n$ is even.

Then by (1) $f$ is
locally decreasing on $p(M)/E_k$ for every odd $k\le n$ and $f$ is locally increasing
on $p(M)/E_k$ for every even $k\le n$.. Take an arbitrary $\gamma\in p(M)$.
Then there exist $a_1, a_2\in p(M)$ such that $a_1<a_2$, $E_2(a_1, a_2)$, $\neg E_1(a_1, a_2)$, $f(a_2)<\gamma<f(a_1)$.
Then we assert that the following convex sets are contained in $F(\gamma, M)$:
the sets definable by $\neg E_n(x,a_2)\land a_2<x\le \gamma$, $E_k(x,a_2)\land \neg E_{k-1}(x,a_2)\land x<a_2$
for every even $1< k\le n$,
$E_k(x,a_2)\land \neg E_{k-1}(x,a_2)\land x>a_2$ for every odd $2< k\le n$, and
a nonempty proper convex subset of the set definable by $E_2(x,a_2)\land \neg E_1(x,a_2)\land x<a_2$.
Observe that these convex sets are definably separable. For example,
$$\neg E_n(x,a_2)\land a_2<x\le \gamma \mbox{ and } E_n(x,a_2)\land \neg E_{n-1}(x,a_2)\land x<a_2$$
are separable by the formula $E_n(x,a_2)\land \neg E_{n-1}(x,a_2)\land x\ge a_2$.
Thus, $F(\gamma, M)$ consists of $n$ convex sets in $p(M)$.

{\it Case B.} $n$ is odd. This case considered similarly. \end{proof}

\begin{theorem}\label{th_neg} Let $T$ be a weakly o-minimal theory, $M\models T$, $A\subseteq M$,
$p\in S_1(A)$ non-algebraic, $F(x,y)$ a $p$-preserving convex-to-right formula that is not equivalence-generating.
Then $$G(x,y):=\exists t [F(y,t)\land t\le x\le y]$$ is a $p$-preserving convex-to-left formula
that is also not equivalence-ge\-ne\-ra\-ting.
\end{theorem}

\begin{proof} Consider $f(y):=\sup F(M,y)$.
By Theorems \ref{f_p} and \ref{dep_th} either $f$ is locally monotonic of depth $n$
for some $n\in\omega$ or $f$ is locally constant of depth $n$ for some natural $n\ge 1$.

{\it Case 1.} $f$ is locally monotonic of depth $n$ for some $n\in\omega$.

By Proposition \ref{prop_locmon} $F(\gamma, M)$ consists of $n+1$ convex sets in $p(M)$
being to the left from $\gamma$ and including the element $\gamma$. Then obviously
$G(M,\gamma)\cap p(M)$ is convex, $\gamma$ is the right endpoint of $G(M,\gamma)\cap p(M)$
and $\gamma\in G(M,\gamma)$, whence we obtain that $G(x,y)$ is a $p$-preserving convex-to-left formula.
Since $f$ is strictly increasing on $p(M)/E_n$, we obatin that $G(x,y)$ is not equivalence-generating.

{\it Case 2.} $f$ is locally constant of depth $n$ for some natural $n\ge 1$.

By Proposition \ref{prop_locon} $F(\gamma, M)$ consists of $n$ convex sets in $p(M)$
being to the left from $\gamma$ and including the element $\gamma$.
The remaining reasons are similar as in Case 1.
\end{proof}

\bigskip
\begin{thebibliography}{1} % Ссылки на литературу приводятся только на английском языке (даже если у источника нет англоязычной версии).


\bibitem{mms} H.D.~Macpherson, D.~Marker, and C.~Steinhorn, {\it Weakly
o-minimal structures and real closed fields}, Transactions
of The American Mathematical Society, \textbf{352}:12 (2000), 5435--5483.

\bibitem{bbs4} B.S.~Baizhanov,  {\it Orthogonality of one-types in weakly
o-minimal theo\-ries}, Algebra and Model Theory II,
(A.G.~Pinus and K.N.~Ponomaryov, editors), Novosibirsk State
Technical University, 1999, 3--28.

\bibitem{bk} B.S.~Baizhanov, B.Sh.~Kulpeshov, {\it On behaviour of $2$-formulas
in weakly o-minimal theories}, Mathematical Logic in Asia, Proceedings of the 9th Asian
Logic Conference (editors  S.~Goncharov, R.~Downey, H.~Ono), Singapore, World Scientific, 2006, 31--40.

\bibitem{jms} B.Sh.~Kulpeshov, {\it Countably categorical quite o-minimal theories}, Journal
of Ma\-the\-ma\-ti\-cal Sciences, \textbf{188}:4 (2013), 387--397.

\bibitem{smj_2021} B.Sh.~Kulpeshov, \href{https://doi.org/10.1134/S0037446621060082} {\it A criterion for binarity of almost $\omega$-categorical
weakly o-minimal theories}, Siberian Mathematical Journal, \textbf{62}:2 (2021), 1063--1075.

\bibitem{vvv} V.V.~Verbovskiy,   {\it On depth of functions of weakly o-minimal structures and an
example of a weakly o-minimal structure without a weakly o-minimal theory}, Pro\-ceedings of In\-for\-ma\-tics and Control
Problems In\-sti\-tute, 1996, 207--216.

\bibitem{vvv2} V.V.~Verbovskiy,  {\it On formula depth  of  weakly
o-minimal structures},   Algebra   and   Model   Theory,
(A.G.~Pinus and K.N.~Ponomaryov,  editors),  Novosibirsk,  1997,
 209--223.

\bibitem{k_2015} B.Sh.~Kulpeshov, {\it Convexity rank and orthogonality in weakly o-minimal
theories}, News of the National Academy of Sciences of the Republic of Kazakhstan,
physical and mathematical series, \textbf{227} (2003), 26--31.

\bibitem{k1} B.Sh.~Kulpeshov, \href{https://doi.org/10.2307/2586664} {\it Weakly o-minimal structures and some
of their pro\-per\-ties}, The Journal of Symbolic Logic, \textbf{63}:4 (1998), 1511--1528.

\bibitem{KS_2017} B.Sh.~Kulpeshov, S.V.~Sudoplatov, \href{https://doi.org/10.1016/j.apal.2016.09.002} {\it Vaught's conjecture for quite o-minimal theories},
Annals of Pure and Applied Logic, \textbf{168}:1 (2017), 129--149.

\bibitem{ABKZ2018} A. Alibek, B.S. Baizhanov, B.Sh. Kulpeshov, T.S. Zambarnaya,
\href{https://doi.org/10.1016/j.apal.2018.06.003} {\it Vaught's conjecture for weakly o-minimal theories of convexity rank 1},
Annals of Pure and Applied Logic, \textbf{169}:11 (2018), 1190--1209.

\bibitem{Izv_2020} B.Sh.~Kulpeshov, \href{https://doi.org/10.1070/IM8894} {\it Vaught's conjecture for weakly o-minimal
theories of finite convexity rank}, Izves\-tiya: Mathematics, \textbf{84}:2 (2020), 324--347.
\end{thebibliography}
\end{document}
