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\begin{document}
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\title[Algebras of binary formulas]{\large Algebras of binary formulas for $\aleph_0$-categorical weakly circularly minimal theories: piecewise monotonic case}


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







\thanks{\sc Kulpeshov, B.Sh.,
Algebras of binary formulas for $\aleph_0$-categorical weakly circularly minimal theories: piecewise monotonic case}
\thanks{\copyright \ 2023 Kulpeshov B.Sh.}
\thanks{\rm The 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:} Algebras of binary isolating formulas are described for $\aleph_0$-categorical 1-transitive
non-primitive weakly circularly
minimal theories of convexity rank greater than 1 having a non-trivial piecewise (non-strictly) monotonic function.\medskip

\noindent{\bf Keywords:} weak circular mi\-ni\-ma\-li\-ty, algebra of binary formulas,
 $\aleph_0$-categorical theory,  circularly ordered structure, convexity rank.
 \end{quote}
}

\bigskip

\section{Preliminaries}
Algebras of binary formulas are a tool for describing
relationships between elements of the sets of realizations of a type at the
binary level with respect to the superposition of binary definable
sets.  A {\it binary isolating formula} is a formula of the form $\varphi(x,y)$
such that for some parameter $a$ the formula $\varphi(a,y)$
isolates a complete type in $S(\{a\})$. The concepts and notations
related to these algebras can be found in the papers
\cite{Sud2018, ShS}. In recent years, algebras of binary formulas
have been studied intensively and have been continued in the works
\cite{Su122}--\cite{AKS_2021}.

Let $L$ be a countable first-order language. Throughout we
consider $L$-structures and assume that $L$ contains a ternary
relational symbol $K$, inter\-pre\-ted as a circular order in these
structures (unless otherwise stated).

Let $\mathcal{M} = \langle M, \le\rangle$ be a linearly ordered
set. If we connect two endpoints of $\mathcal{M}$ (possibly,
$-\infty$ and $+\infty$), then we obtain a circular order. More
formally, the {\it circular order} is described by a ternary
relation $K$ satisfying the following conditions:

(co1) $\forall x\forall y \forall z (K(x,y,z)\to K(y,z,x))$;

(co2) $\forall x\forall y \forall z (K(x,y,z)\land K(y,x,z)
 \Leftrightarrow x=y \lor y=z \lor z=x)$;

 (co3) $\forall x\forall y \forall z(K(x,y,z)\to \forall t[K(x,y,t)
 \lor K(t,y,z)])$;

 (co4) $\forall x\forall y \forall z (K(x,y,z)\lor K(y,x,z))$.

Sometimes we will identify $\mathcal{M}$ and the universe $M$ if a
linear/circular order is fixed.

The notion of {\it weak circular minimality} was studied initially
in \cite{km}. Let $A\subseteq M$, where $\mathcal{M}$ is a
circularly ordered structure. The set $A$ is called {\it convex}
if for any $a, b\in A$ the following property is satisfied:  for
any $c\in M$ with $K(a,c,b)$, $c\in A$ holds, or for any $c\in M$
with $K(b,c,a)$, $c\in A$ holds. A {\it weakly circularly
minimal structure} is a circularly ordered structure
$\mathcal{M}=\langle M, K,\ldots\rangle$ such that any definable
(with parameters) subset of $M$ is a union of finitely many convex
sets in $\mathcal{M}$. Recall \cite{ms} that such a structure
$\mathcal{M}$ is called {\it circularly minimal} if any definable
(with parameters) of $M$ is a union of finitely many intervals and
points in $\mathcal{M}$. Clearly, the weak circular minimality is
a generalization of circular minimality. Notice also that any
weakly o-minimal structure is weakly circular minimal. The
converse, in general, fails. The study of weakly circularly
minimal structures was continued in the papers
\cite{Kul06}--\cite{MN_2021}.

Let $\mathcal{M}$ be an $\aleph_0$-categorical weakly circularly minimal
structure, $G:={\rm Aut}(\mathcal{M})$. Following the standard
group theory terminology, the group $G$ is called {\it
$k$-transitive} if for any pairwise distinct $a_1, a_2, \ldots,
a_k\in M$ and pairwise distinct $b_1, b_2, \ldots, b_k\in M$ there
exists $g \in G$ such that $g(a_1)=b_1, g(a_2)=b_2, \ldots,
g(a_k)=b_k$. A {\it congruence} on $\mathcal{M}$ is an arbitrary
$G$-invariant equivalence relation on $\mathcal{M}$. The group $G$
is called {\it primitive} if $G$ is $1$-transitive and there are
no non-trivial proper congruences on $\mathcal{M}$.

Let $\mathcal{M}$, $\mathcal{N}$ be cir\-cu\-lar\-ly ordered structures.
The {\it $2$-reduct} of $\mathcal{M}$ is a cir\-cu\-lar\-ly ordered
structure with the same universe of $\mathcal{M}$ and consisting
of predicates for each $\emptyset$-definable relation on
$\mathcal{M}$ of arity $\leq 2$ as well as of the ternary
predicate $K$ for the circular order, but does not have other
predicates of arities more than two. We say that the structure
$\mathcal{M}$ is {\it isomorphic} to $\mathcal{N}$ {\it up to
binarity} or {\it binarily isomorphic to} $\mathcal{N}$ if a
$2$-reduct of $\mathcal{M}$ is isomorphic to a $2$-reduct of
$\mathcal{N}$.
\vskip 3mm\noindent
{\bf Notation.}

(1)   $K_0(x,y,z):=K(x,y,z)\land y\ne x \land y \ne z \land x\neq
z$.

 (2) $K(u_1,\ldots,u_n)$ denotes a formula saying that all subtuples of the tuple
$\langle u_1, \ldots, u_n\rangle$ having the length 3 (in
ascending order) satisfy $K$; similar notations are used for
$K_0$.

(3) Let $A, B, C$ be disjoint convex subsets of a circularly
ordered structure $\mathcal{M}$. We write $K(A,B,C)$ if for any
$a,b,c\in M$ with $a\in A$, $b\in B$, $c\in C$ we have $K(a,b,c)$.
We extend naturally that notation using, for instance, the
notation $K_0(A,d,B,C)$ if $d\not\in A\cup B\cup C$ and
$K_0(A,d,B)\wedge K_0(d,B,C)$ holds.
\vskip 3mm

The following definition can be used in a circular ordered structure as well.
\begin{definition}\rm \cite{k1}, \cite{SMJ2021}
Let $T$ be a weakly o-minimal theory, $M$ be a sufficiently sa\-tu\-ra\-ted model of $T$, $A\subseteq M$.
{\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.

\vskip 2mm
Let $f: I\to M$ be an $\emptyset$-definable function with $Dom(f)=I\subseteq M$,
where $I$ is an open convex set.
We say that  $f$ is {\it monotonic-to-right (left) on $I$} if it preserves (reverses) the relation
$K_0$, i.e. for any $a,b,c \in I$ such that $K_0(a,b,c)$ we have
$K_0(f(a),f(b),f(c))$ ($K_0(f(c),f(b),f(a))$).
We also say that $f$
is {\it piecewise monotonic-to-right (left) on $M$} if there exists an
$\emptyset$-definable non-trivial equivalence relation $E(x,y)$ partitioning $M$
into finitely many infinite convex classes so that $f$ is mo\-no\-to\-nic-to-right on each $E$-class
and $f$ is not monotonic-to-left (right) on $M/E$, where by $M/E$ we denote the set of
representatives of $E$-classes in $M$.
\vskip 2mm\noindent
{\bf Example. }\rm \cite{Kul06} Let $M:=\langle M, K, E^2, f^1\rangle$ be a circularly ordered structure,
where $M$ is a disjoint union of $\mathbb{Q}_1$, $\mathbb{Q}_2$, $\ldots$, $\mathbb{Q}_6$, where
$\mathbb{Q}_i$ is a copy of the ordering of rational numbers $\mathbb{Q}$. The symbol $E$ interprets
an equivalence relation on $M$ as follows: $E(a,b)$ iff there is $1\le i\le 6$ with $a,b\in\mathbb{Q}_i$.
The symbol $f$ interprets a function on $M$ as follows: $f(\mathbb{Q}_i)=\mathbb{Q}_{i+3}$ for each $1\le i\le 3$,
$f(\mathbb{Q}_j)=\mathbb{Q}_{j-3}$ for each $4\le j\le 6$, and $f(q)=-q$ for all $q\in \mathbb{Q}$.

It can be proved that $M$ is an $\aleph_0$-categorical 1-transitive weakly circularly minimal structure,
$f$ is a bijection on $M$ so that $f^2(a)=a$ for all $a\in M$, $f$ is monotonic-to-left on each $E$-class and
$f$ is monotonic-to-right on $M/E$, i.e. $f$ is piecewise monotonic-to-left on $M$.


\begin{lemma} {\rm \cite{Kul06}} Let $M$ be an $\aleph_0$-categorical 1-transitive weakly circularly minimal structure,
$f$ be an $\emptyset$-definable function on $M$. Then $f$ cannot be piecewise monotonic-to-right on $M$.
\end{lemma}


The following theorem  characterizes
$\aleph_0$--categorical 1-transitive non-pri\-mi\-tive weakly cir\-cu\-lar\-ly minimal structures
of convexity rank greater  than 1 having  a non-trivial piecewise (non-strictly) monotonic  function up to binarity:


\begin{theorem} \label{th1_rc2} {\rm \cite{Kul09}} (piecewise monotonic case) Let $M$ be an
$\aleph_0$--categorical 1-transitive non-primitive weakly circularly minimal structure of
convexity rank greater than  1 having a non-trivial piecewise (non-strictly) monotonic
function so that  $dcl(a)\ne\{a\}$ for some $a\in M$. Then $M$
is isomorphic up to binarity to  $M_{s,m,k}:=\langle
M$, $K$, $f^1$, $E^2_1$, $\ldots, E^2_s, E^2_{s+1}\rangle$,
where
\begin{itemize}
\item $M$ is a circularly ordered structure, $M$ is densely ordered, $s\ge 1$, $k\ge 2$, $k$ is even,
 $k$ divides $m$, $m\ge 4$;
\item $E_{s+1}$ is an equivalence relation partitioning $M$ into $m$
infinite convex classes without endpoints, for every
$1\le i \le s$ the relation $E_i$ is an equivalence relation partitioning every $E_{i+1}$--class
into infinitely many infinite convex $E_i$--subclasses without endpoints so that the induced order
on  $E_i$--subclasses is dense without endpoints;
\item $f$ is a bijection on $M$ so that $f^k(a)=a$ for any $a\in M$,
for every $1\le i\le s+1$ $f(E_i(M,a))=E_i(M,f(a))$ and $\neg
E_i(a,f(a))$, and  $f$ is piecewise monotonic-to-left on $M$, i.e.  $f$ is monotonic-to-left
on each $E_{s+1}$--class and $f$
is monotonic-to-right on $M/E_{s+1}$.
\end{itemize}


\end{theorem}

In \cite{EKS_2021} algebras of binary isolating
formulas are described for $\aleph_0$-ca\-te\-go\-ri\-cal weakly
cir\-cu\-lar\-ly minimal theories with a primitive automorphism group.
In \cite{KS_2022} algebras of binary isolating formulas
are described for $\aleph_0$-categorical weakly circularly minimal theories of
con\-ve\-xi\-ty rank 1 with a 1-transitive non-primitive automorphism
group and a non-trivial definable closure. Here we describe algebras of binary isolating formulas
 for $\aleph_0$-categorical weakly circularly minimal theories of
convexity rank greater than 1 with a 1-tran\-si\-tive non-primitive automorphism
group and having a non-trivial piecewise (non-strictly) monotonic function.

\section{Results}



\begin{definition}\rm \cite{Sud2018} Let $p\in S_1(\emptyset)$ be non-algebraic.
The algebra $\mathcal{P}_{\nu(p)}$ is said to be {\it de\-ter\-mi\-nis\-tic} if $u_1\cdot u_2$
is a singleton for any labels $u_1, u_2\in \rho_{\nu(p)}$.
\end{definition}

Generalizing the last definition, we say that the algebra $\mathcal{P}_{\nu(p)}$ is {\it $m$-de\-ter\-mi\-nis\-tic}
if the product $u_1\cdot u_2$ consists of at most $m$ elements for any labels $u_1, u_2\in \rho_{\nu(p)}$. We also say
that an $m$-deterministic algebra $\mathcal{P}_{\nu(p)}$ is {\it strictly $m$-deterministic} if it is not $(m-1)$-deterministic.

\vskip 3mm\noindent
{\bf Example.} \rm Consider the structure $M_{1,4,2}:=\langle M, K^3, f^1, E^2_1, E^2_2\rangle$ from Theo\-rem \ref{th1_rc2},
where $f$ is piecewise monotonic-to-left on $M$, $E_1$ is an equivalence relation partitioning $M$ into
infinitely many infinite convex classes, $E_2$ is an equivalence relation partitioning $M$ into
four infinite convex classes.

We assert that  $Th(M_{1,4,2})$ has twelve binary isolating formulas:
$$\theta_0(x,y):=x=y, \theta_1(x,y):=K_0(x,y,f(x))\land E_1(x,y),$$
$$\theta_2(x,y):=K_0(x,y,f(x))\land \neg E_1(x,y)\land E_2(x,y),$$
$$\theta_3(x,y):=K_0(x,y,f(x))\land \neg E_2(x,y)\land \neg E_2(f(x),y),$$
$$\theta_4(x,y):=K_0(x,y,f(x))\land \neg E_1(f(x),y)\land E_2(f(x),y),$$
$$\theta_5(x,y):=K_0(x,y,f(x))\land E_1(f(x),y),$$
$$\theta_6(x,y):=f(x)=y, \theta_7(x,y):=K_0(f(x),y,x)\land E_1(f(x),y),$$
$$\theta_8(x,y):=K_0(f(x),y,x)\land \neg E_1(f(x),y)\land E_2(f(x),y),$$
$$\theta_9(x,y):=K_0(f(x),y,x)\land \neg E_2(f(x),y)\land \neg E_2(x,y),$$
$$\theta_{10}(x,y):=K_0(f(x),y,x)\land \neg E_1(x,y)\land E_2(x,y),$$
$$\theta_{11}(x,y):=K_0(f(x),y,x)\land E_1(x,y),$$

and $$K_0(\theta_i(a,M),\theta_{i+1}(a,M),\theta_{i+2}(a,M)), \mbox{ where } 0\le i\le 9,$$
$$ K_0(\theta_{10}(a,M),\theta_{11}(a,M),\theta_0(a,M)), \quad K_0(\theta_{11}(a,M),\theta_0(a,M),\theta_1(a,M))$$
hold for any $a\in M$.

Define labels for these formulas as follows:
$$\mbox{label } k \mbox{ for } \theta_k(x,y) \mbox{ where } 0\le k\le 11.$$

It easy to check that for the algebra $\mathfrak{P}_{M_{1,4,2}}$ the Cayley table has the following form:

\begin{center}
\tiny{\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
 $\cdot$ & $0$  & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$\\
 \hline
$0$ & $\{0\}$ & $\{1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{5\}$ & $\{6\}$ & $\{7\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{11\}$\\
 \hline
$1$ & $\{1\}$ & $\{1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{5\}$ & $\{5\}$ & $\{5,6,7\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{11,0,1\}$\\
 \hline
$2$ & $\{2\}$ & $\{2\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{4\}$ & $\{4\}$ & $\{4\}$ & $\ldots$ & $\{9\}$ & $\{10,11,0,1,2\}$ & $\{2\}$\\ %\{4,5,6,7,8\}
 \hline
$3$ & $\{3\}$ & $\{3\}$ & $\{3\}$ & $\ldots$ & $\{9\}$ & $\{9\}$ & $\{9\}$ & $\{9\}$ & $\{9\}$ & $\ldots$ & $\{3\}$ & $\{3\}$\\%\{4,5,6,7,8\}, \{10,11,0,1,2\}
 \hline
$4$ & $\{4\}$ & $\{4\}$ & $\ldots$ & $\{9\}$ & $\ldots$ & $\{2\}$ & $\{2\}$ & $\{2\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{4\}$\\%\{4,5,6,7,8\}, \{10,11,0,1,2\}
 \hline
$5$ & $\{5\}$ & $\{5,6,7\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{11,0,1\}$ & $\{1\}$ & $\{1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{5\}$\\
 \hline
$6$ & $\{6\}$ & $\{7\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$& $\{11\}$ & $\{0\}$ & $\{1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{5\}$\\
 \hline
$7$ & $\{7\}$ & $\{7\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{11\}$ & $\{11\}$ & $\{11,0,1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{5,6,7\}$\\
 \hline
$8$ & $\{8\}$ & $\{8\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{10\}$ & $\{10\}$ & $\{10\}$ & $\ldots$ & $\{3\}$ & $\{4,5,6,7,8\}$ & $\{8\}$\\% \{10,11,0,1,2\}
 \hline
$9$ & $\{9\}$ & $\{9\}$ & $\{9\}$ & $\ldots$ & $\{3\}$ & $\{3\}$ & $\{3\}$ & $\{3\}$ & $\{3\}$ & $\ldots$ & $\{9\}$ & $\{9\}$\\%\{10,11,0,1,2\}, \{4,5,6,7,8\}
 \hline
$10$ & $\{10\}$ & $\{10\}$ & $\ldots$ & $\{3\}$ & $\ldots$ & $\{8\}$ & $\{8\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{10\}$ & $\{10\}$\\%\{10,11,0,1,2\}, \{4,5,6,7,8\}
 \hline
$11$ & $\{11\}$ & $\{11,0,1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$ & $\{5,6,7\}$ & $\{7\}$ & $\{7\}$ & $\{8\}$ & $\{9\}$ & $\{10\}$ & $\{11\}$\\
 \hline
\end{tabular}}
\end{center}

By the Cayley table the algebra $\mathfrak{P}_{M_{1,4,2}}$ is not commutative.


\begin{theorem}\label{algebra_wcm4} The algebra $\mathfrak{P}_{M_{s,m,k}}$
of binary isolating formulas having a piecewise mo\-no\-to\-nic-to-left function on $M$
has $2k(s+1)+m$ labels, is strictly $(2s+3)$-deterministic and is not commutative.
\end{theorem}

\begin{proof}

Indeed, since $f^k(a)=a$, we have the following isolating formulas:
$$f^l(x)=y \mbox{ for every }  0\le l\le k-1.$$

Since for every $1\le i \le s$ the relation $E_i$ is an equivalence relation partitioning every
$E_{i+1}$--class into infinitely many infinite convex $E_i$--sub\-clas\-ses without endpoints so that the induced order on
$E_i$--subclasses is dense without endpoints, we obtain the following binary isolating formulas:
$$K_0(f^l(x),y,f^{l+1}(x))\land E_1(f^l(x),y), \mbox{ where } 0\le l\le k-1,$$
$$K_0(f^l(x),y,f^{l+1}(x))\land \neg E_j(f^l(x),y)\land E_{j+1}(f^l(x),y),$$ $$\mbox{ where } 0\le l\le k-1, 1\le j\le s-1,$$
$$K_0(f^l(x),y,f^{l+1}(x))\land \neg E_s(f^l(x),y)\land \neg E_s(f^{l+1}(x),y), \mbox{ where } 0\le l\le k-1,$$
$$K_0(f^l(x),y,f^{l+1}(x))\land \neg E_j(f^{l+1}(x),y)\land E_{j+1}(f^{l+1}(x),y),$$ $$\mbox{ where } 0\le l\le k-1, 1\le j\le s-1,$$
$$K_0(f^l(x),y,f^{l+1}(x))\land E_1(f^{l+1}(x),y), \mbox{ where } 0\le l\le k-1.$$

Since in this structure there exists additionally an equivalence relation $E_{s+1}(x,y)$
par\-ti\-tio\-ning $M$ into $m$ infinite convex classes, additionally the following binary isolating formulas appear:
$$K_0(f^l(x),y,f^{l+1}(x))\land \neg E_s(f^l(x),y)\land E_{s+1}(f^l(x),y),  $$
$$K_0(f^l(x),y,f^{l+1}(x))\land \neg E_s(f^{l+1}(x),y)\land E_{s+1}(f^{l+1}(x),y),$$
where $0\le l\le k-1$.

Also, the formulas  $\theta^{l,i}(x,y)$ containing the conjunctive term $K_0(f^l(x)$, $y$, $f^{l+1}(x))$ and
extracting the $i$-th $E_{s+1}$-class to the right of $E_{s+1}$-class containing $f^l(x)$ for some
$1\le i\le m/k-1$ (here also $0\le l\le k-1$) will be binary isolating formulas. For example,
the formula $\theta^{l,1}(x,y)$ has the following form:
$$\theta^{l,1}(x,y):= K_0(f^l(x),y,f^{l+1}(x))\land\neg E_{s+1}(f^l(x),y)\land$$ $$\forall t[K_0(f^l(x), t,y)\land \neg E_{s+1}(t,y)\to E_{s+1}(f^l(x),t)].$$

Thus, we obtain $k+k+2k(s-1)+k+2k+k(m/k-1)=2k(s+1)+m$ binary isolating formulas.

The formulas $$\exists t [f^l(x)=t\land K_0(f^l(t),y,f^{l+1}(t))\land E_1(f^l(t),y)]$$ and
$$\exists t[K_0(f^l(x),t,f^{l+1}(x))\land E_1(f^l(x),t) \land f^l(t)=y],$$ where $0\le l\le k-1$,
uniquely determine the formula $$K_0(f^{2l \mbox{mod } k}(x),y,f^{2l+1 \mbox{mod } k}(x))\land E_1(f^{2l \mbox{mod } k}(x),y).$$


The formula $$\exists t [K_0(x,t,f(x))\land E_1(x,t)\land K_0(f(t), y, f^2(t))\land E_1(f(t),y)]$$
is compatible with the following formulas:
$$K_0(x,y,f(x))\land E_1(f(x),y),$$ $$f(x)=y,$$ $$K_0(f(x), y,f^2(x))\land E_1(f(x),y).$$

While the formula
$$\exists t [K_0(f(x),t,f^2(x))\land E_1(f(x),t)\land K_0(t, y, f(t))\land E_1(t,y)]$$
uniquely determines the formula $K_0(f(x),y,f^2(x))\land E_1(f(x),y)$.
Con\-se\-quent\-ly, the algebra $\mathfrak{P}_{M_{s,m,k}}$ is not commutative.

And in general we consider the formula $$\exists t [K_0(f^{l_1}(x),t,f^{l_1+1}(x))\land E_1(f^{l_1}(x),t)\land$$ $$ K_0(f^{l_2}(t),y,f^{l_2+1}(t))\land E_1(f^{l_2}(t),y)],$$
where $0\le l_1, l_2\le k-1$.
Such a formula for even $l_2$ uniquely determines the formula
$$K_0(f^{l_1+l_2 \mbox{(mod } k)}(x), y, f^{l_1+l_2+1 \mbox{(mod } k)}(x))\land E_1(f^{l_1+l_2 \mbox{(mod } k)}(x),y),$$
and for odd $l_2$ it is compatible with the following three formulas:
$$K_0(f^{l_1+l_2 \mbox{(mod } k)}(x),y,f^{l_1+l_2+1 \mbox{(mod } k)}(x))\land E_1(f^{l_1+l_2 \mbox{(mod } k)}(x),y),$$
$$f^{l_1+l_2 \mbox{(mod } k)}(x)=y,$$ $$K_0(f^{l_1+l_2-1 \mbox{(mod } k)}(x), y,f^{l_1+l_2 \mbox{(mod } k)}(x))\land E_1(f^{l_1+l_2 \mbox{(mod } k)}(x),y).$$

Further consider the formula
$$\exists t [K_0(f^{l_1}(x),t,f^{l_1+1}(x))\land E_1(f^{l_1}(x),t)\land K_0(f^{l_2}(t),y,f^{l_2+1}(t))$$ $$\land \neg E_j(f^{l_2}(t),y)\land E_{j+1}(f^{l_2}(t),y)],$$
where $0\le l_1, l_2\le k-1$, $1\le j\le s$. It uniquely determines the formula
$$K_0(f^{l_1+l_2 \mbox{(mod } k)}(x), y, f^{l_1+l_2+1 \mbox{(mod } k)}(x))\land \neg E_j(f^{l_1+l_2 \mbox{(mod } k)}(x),y)$$
$$\land E_{j+1}(f^{l_1+l_2 \mbox{(mod } k)}(x),y).$$
On the other hand, the formula
$$\exists t [K_0(f^{l_2}(x),t,f^{l_2+1}(x)) \land \neg E_j(f^{l_2}(x),t)\land E_{j+1}(f^{l_2}(x),t)$$ $$\land K_0(f^{l_1}(t),y,f^{l_1+1}(t))\land E_1(f^{l_1}(t),y)]$$
uniquely determines the formula
$$K_0(f^{l_1+l_2-1 \mbox{(mod } k)}(x), y, f^{l_1+l_2 \mbox{(mod } k)}(x))\land \neg E_j(f^{l_1+l_2 \mbox{(mod } k)}(x),y)$$
$$\land E_{j+1}(f^{l_1+l_2 \mbox{(mod } k)}(x),y).$$

Consider now the formulas $\theta^{l_1, i}(x,y)$ and $\theta^{l_2, j}(x,y)$ for arbitrary $0\le l_1, l_2\le k-1$, $1\le i, j\le m/k-1$.
If $i+j \mbox{ (mod } m/k)\ne 0$, it easy to check that the formulas
$$\exists t[\theta^{l_1, i}(x,t)\land \theta^{l_2, j}(t,y)] \mbox{ and } \exists t [\theta^{l_2, j}(x,t)\land \theta^{l_1, i}(t,y)]$$
uniquely determine the formula $\theta^{l_1+l_2 \mbox{(mod } k), i+j\mbox{(mod } m/k)}(x,y)$.

If $i+j \mbox{ (mod } m/k)= 0$, these formulas are compatible with the following $2s+3$ formulas:
$$f^{l_1+l_2+1 \mbox{(mod } k)}(x)=y,$$
$$K_0(f^{l_1+l_2 \mbox{(mod } k)}(x), y, f^{l_1+l_2+1 \mbox{(mod } k)}(x))\land E_1(f^{l_1+l_2+1 \mbox{(mod } k)}(x),y),$$
$$K_0(f^{l_1+l_2 \mbox{(mod } k)}(x), y, f^{l_1+l_2+1 \mbox{(mod } k)}(x))\land \neg E_j(f^{l_1+l_2+1 \mbox{(mod } k)}(x),y)$$
$$\land E_{j+1}(f^{l_1+l_2+1 \mbox{(mod } k)}(x),y),\quad 1\le j\le s,$$
$$K_0(f^{l_1+l_2+1 \mbox{(mod } k)}(x), y, f^{l_1+l_2+2 \mbox{(mod } k)}(x))\land E_1(f^{l_1+l_2+1 \mbox{(mod } k)}(x),y),$$
$$K_0(f^{l_1+l_2+1 \mbox{(mod } k)}(x), y, f^{l_1+l_2+2 \mbox{(mod } k)}(x))\land \neg E_j(f^{l_1+l_2+1 \mbox{(mod } k)}(x),y)$$
$$\land E_{j+1}(f^{l_1+l_2+1 \mbox{(mod } k)}(x),y),\quad 1\le j\le s.$$

Thus, the algebra $\mathfrak{P}_{M_{s,m,k}}$ is strictly $(2s+3)$-deterministic.
\end{proof}





\bigskip
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\medskip

Beibut Shaiykovich Kulpeshov

Institute of Mathematics and Mathematical Modeling,

Shevchenko street, 28,

050010, Almaty, Kazakhstan

E-mail: kulpesh@mail.ru

 Kazakh British Technical University,

Tole bi street, 59,

050000, Almaty, Kazakhstan

E-mail: b.kulpeshov@kbtu.kz

Novosibirsk State Technical University,

K. Marx avenue, 20,

630073, Novosibirsk, Russia

E-mail: kulpeshov@corp.nstu.ru

\end{document}
