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\title[\sc On algebras of binary formulas ]{\large On algebras of binary formulas for weakly circularly minimal theories of finite convexity rank}
\author[B.Sh.~Kulpeshov and S.V.~Sudoplatov]{{\bf \large \textsc{B.Sh.~Kulpeshov}\protect\orcidicon{0000-0002-4242-0463}, \textsc{S.V.~Sudoplatov}}\protect\orcidicon{0000-0002-3268-9389}\\ \\
\textit{Представлено} \textsc{П.П. Петровым}
%\textit{(заполняется редактором)}
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\address{Beibut Shaiykovich Kulpeshov
\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}}%


% .
%, , , ; .

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



\thanks{\sc Kulpeshov, B.Sh., Sudoplatov, S.V.,
On algebras of binary formulas for weakly circularly minimal theories of finite convexity rank}
\thanks{\copyright \ 2025 Kulpeshov, B.Sh., Sudoplatov, S.V}
\thanks{\rm This work was supported by Science Committee of
Ministry of Science and Higher Education of the Republic of
Kazakhstan, Grant No.~BR20281002, and it was carried out in the
framework of the State Contract of Sobolev Institute of
Mathematics, Project No.~FWNF-2022-0012.}
\thanks{\it Поступила 1 января 2023 г., опубликована 31 декабря 2023 г.}%


\semrtop \vspace{1cm}
\maketitle {\small
\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 finite convexity rank with a trivial definable
closure having a monotonic-to-right function to the definable completion of a structure and
not having a non-trivial equivalence relation partitioning the universe of a structure into
finitely many convex classes. \medskip

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

\bigskip

\section{Preliminaries}
\noindent

Algebras of binary formulas are a tool for describing
relationships between elements of the sets of realizations of an
one-type at the binary level with respect to the superposition of
binary definable sets. These algebras, as natural derivative
structures with respect to initial ones, reflecting binary links,
allow to clarify the structural behavior and to classify both the
struc\-tu\-res and their elementary theories. Essential structural
properties of algebras of binary formulas include the (right)
associativity, the (partial) com\-mu\-ta\-ti\-vi\-ty, the (almost)
determinacy, the (almost) absorbing, etc. These properties can be
observed on a base of Cayley tables explicitly defining algebras.

The concepts and notations related to algebras of binary formulas
can be found in \cite{ShS, Sud2018}. 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 $S(\{a\})$.  Given a complete $1$-type $p(x)$ over
$\emptyset$ realized in a structure $M$, one considers the binary
isolating formulas $\varphi(x,y)$. Then one considers an
equivalence relation on such formulas (logical equivalence of
$\varphi(a,y)$ and $\psi(a,y)$ for an element $a$ realizing
$p(x))$. The classes of this equivalence are then embedded in an
algebra of labels with a composition-like operation. Composing two
(equivalence classes of) binary isolating formulas $\varphi(x,y)$
and $\psi(y,z)$ results in the set of all (classes of) isolating
formulas $\chi(x,z)$ such that $\chi(a,z)$ implies $\exists
y(\varphi(a,y)\wedge\psi(y,z))$. Thus the operator collecting
labels for $\chi$ with respect to labels for $\varphi$ and $\psi$
generates the algebra on the Boolean of labels, which is the
multi-algebra on the set of labels related to the type $p(x)$.

In recent years, algebras of binary formulas have been studied
intensively for various classes of theories and have been
continued in \cite{AKS_2021}--\cite{SMJ_2024}.

In the present paper, algebras of binary isolating formulas are
described for the natural class of $\aleph_0$-categorical
1-transitive non-primitive weakly circularly minimal theories of
finite convexity rank with a trivial definable closure having a
monotonic-to-right function to the definable completion of a
structure and not having a non-trivial equivalence relation
partitioning the universe of a structure into finitely many convex
classes. The main result of the paper is the explicit
characterization of the algebras of binary isolating formulas
under the conditions above. In particular, a sharp bound is
obtained for the degree of determinacy, i.e. the maximal size of a
product of two labels.

\medskip
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 $M = \langle M, \le\rangle$ be a linearly ordered
set. If we connect two endpoints of $M$ (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) $\forall x\forall y \forall z (K(x,y,z)\lor K(y,x,z))$.



The following observation relates linear  and circular orders.
 \begin{fact}{\rm \cite{bmmn}} \label{compat}
(i) If $\langle M,\le\rangle$ is a linear ordering and $K$ is the ternary relation
derived from $\le$ by the rule
$$K(x,y,z):\Leftrightarrow (x\le y\le z )\lor (z\le x \le y)\lor (y\le z\le x)$$
then $K$ is a circular order relation on $M$.

(ii) If $\langle N, K\rangle$ is a circular ordering and
$a \in N$, then the relation $\le_a$ defined on $M:= N\setminus \{a\}$ by
the rule
$y\le_a z :\Leftrightarrow K(a, y,z)$
is a linear order.
\end{fact}

Thus, any linearly ordered structure is circularly ordered,
since the relation of circular order is $\emptyset$-definable in an arbitrary linearly ordered structure.
However, the opposite is not true. The following example shows that there exist circularly ordered structures
that are not linearly ordered (in the sense that a linear ordering relation is not
$\emptyset$-definable in an arbitrary circularly ordered structure).

\begin{example}\label{dro} \rm \cite{pcam, dr}
Let $\mathbb{Q}_2^*:=\langle \mathbb{Q}_2, K^3, S^2_0, S^2_1 \rangle$ be a circularly ordered structure,
where $K^3$ is a ternary relation expressing a circular order on $\mathbb{Q}_2$, and
the following conditions hold:

(i) its domain $\mathbb{Q}_2$ is a countable dense subset of the unit circle,
 no two points making the central angle $\pi$;

(ii) for distinct $a,b\in \mathbb{Q}_2$
$$S_0(a,b) \mbox{ iff } 0<\arg(a/b)<\pi,$$
$$S_1(a,b) \mbox{ iff } \pi<\arg(a/b)<2\pi,$$
where $\arg(a/b)$ means the value of the central angle between $a$ and $b$ clock\-wise.

Indeed,  one can check that the linear order relation is not $\emptyset$-definable in this structure.
\end{example}


The notion of {\it weak circular minimality} was introduced
in \cite{KulMac}. Let $A\subseteq M$, where $M$ 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
$M=\langle M, K,\ldots\rangle$ such that any definable
(with parameters) subset of $M$ is a union of finitely many convex
sets in $M$.  The study of weakly circularly
minimal structures was continued in the papers
\cite{Kul06}--\cite{AK_2021}.

Let $M$ be an $\aleph_0$-categorical weakly circularly minimal
structure, $G:={\rm Aut}(M)$. 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 $M$ is an arbitrary
$G$-invariant equivalence relation on $M$. The group $G$
is called {\it primitive} if $G$ is $1$-transitive and there are
no non-trivial proper congruences on $M$.

\begin{notat}\rm    \label{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 $M$. 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.
\end{notat}

We say that $M=\langle M,K^3\rangle$ is a {\it dense} circularly ordered
structure if $M$ is a circularly ordered structure and for any $a,b\in M$
with $a\ne b$ there exists $c\in M$ such that $K_0(a,c,b)$.

Further, we need  the notion of the definable completion of a circularly
ordered structure, introduced in \cite{KulMac}.
Its linear analog was introduced in \cite{mms}. A {\it cut} $C(x)$ in a circularly
ordered structure $M$ is maximal consistent set of formulas of the form $K(a,x,b)$, where $a,b\in M$.
A cut is said to be {\it algebraic} if there exists $c\in M$ that realizes it. Otherwise, such a cut
is said to be {\it non-algebraic}.
Let $C(x)$ be a non-algebraic cut. If there is some $a\in M$ such that
either for all $b\in M$ the formula $K(a,x,b)\in C(x)$, or for all $b\in M$ the formula $K(b,x,a)\in C( x)$,
then $C(x)$ is said to be {\it rational}.
Otherwise, such a cut is said to be {\it irrational}.
A {\it definable cut} in $M$ is a cut $C(x)$ with the following property:
there exist $a,b\in M$ such that $K(a,x,b)\in C(x)$ and the set $\{c\in M\mid K(a,c,b)$ and $ K(a,x,c)\in C(x)\}$ is definable.
The {\it definable completion} $\overline{M}$ of a structure $M$ consists of $M$ together with all definable
cuts in $M$ that are irrational (essentially, $\overline{M}$ consists of endpoints of definable subsets of the structure $M$).



\begin{notat}\rm \cite{KulMac}
Let $F(x,y)$ be an $L$-formula such that $F(M,b)$ is convex
infinite co-infinite for each $b\in M$.
Let $F^\ell(y)$ be the formula saying $y$ is a left endpoint of $F(M,y)$:
$$ \exists z_1 \exists z_2 [K_0(z_1,y,z_2)\land \forall t_1(K(z_1,
t_1,y)\land t_1\ne y \to \neg F(t_1,y))\land$$
$$\forall t_2 (K(y,t_2,z_2)\land t_2\ne y\to F(t_2,y))].$$


We say that $F(x,y)$ is {\em convex-to-right} if
$$M\models \forall y \forall x [F(x,y)\to F^l(y)\land \forall z(K(y,z,x)\to
F(z,y))].$$
If $F_1(x,y), F_2(x,y)$ are arbitrary convex-to-right
formulas we say $F_2$ is {\it bigger than} $F_1$ if there is $a\in
M$ with $F_1(M,a)\subset F_2(M,a)$. If $M$ is 1-transitive and this
holds for some $a$, it holds for all $a$. This gives a total ordering on the
(finite) set of all convex-to-right formulas $F(x,y)$ (viewed up to equivalence modulo
$Th(M)$).

Consider $F(M,a)$ for arbitrary $a\in M$. In general,  $F(M,a)$ has no the right endpoint in $M$.
For example, if $dcl(\{a\})=\{a\}$ holds for some $a\in M$ then for any convex-to-right
formula $F(x,y)$ and any $a\in M$ the formula $F(M,a)$ has no the right endpoint in $M$. We write
$f(y):={\rm rend}\; F(M,y)$, assuming that $f(y)$ is the right endpoint of the set $F(M,y)$ that
lies in general in the definable completion   $\overline{M}$ of  $M$.
Then $f$ is a function mapping $M$ in  $\overline{M}$.
\end{notat}

Let $F(x,y)$ be a convex-to-right formula. We say that
$F(x,y)$ is {\it equi\-va\-lence-generating} if for any $a, b \in M$ such that
 $M\models F(b, a)$ the following holds:
$$M\models \forall x (K(b, x, a)\land x\ne a \to [F(x, a)\leftrightarrow F(x, b)]).$$

\vskip 3mm
Consider the following formula:
$$F(x,y):=y=x\lor S_0(y,x),$$
where the predicate $S_0$ from Example \ref{dro}. It can be easily checked that $F(x,y)$
is a convex-to-right formula that is not equivalence-generating.

\begin{example}\rm Let $M=\langle M,K^3, E^2\rangle$ be a dense circularly ordered
struc\-ture, where $E$ is an equivalence relation partitioning $M$ into infinitely
many con\-vex classes without endpoints.  Consider the following formula:
$$F(x,y):=\exists t [\forall u (K(y,u,t)\to E(y,u))\land K(y,x,t)].$$

Also, it can be easily checked that $F(x,y)$ is a convex-to-right formula that is
equivalence-generating.
\end{example}

\begin{lemma}\label{l104} {\rm \cite{ak}}
Let $M$ be an $\aleph_0$-categorical 1-transitive weakly circularly minimal structure,
$F(x,y)$ be a convex-to-right  formula that is equivalence-generating.
Then $E(x, y) := F(x,y)\lor F(y,x)$ is an equivalence relation partitioning $M$ into infinite
convex classes.
\end{lemma}

\begin{notat}\label{n1}\rm Let $E(x,y)$ be an $\emptyset$--definable equivalence relation
partitioning $M$ into infinite convex classes. Suppose that $y$ lies in $\overline{M}$
(not necessarily in $M$). Then $$E^*(x,y):=\exists y_1\exists y_2[y_1\ne y_2\land
\forall t(K(y_1,t,y_2)\to E(t,x))\land K_0(y_1, y, y_2)].$$
 \end{notat}

\begin{example}\label{ex_induced} \rm
Let $M=\langle \mathbb{Q} \times \mathbb{Q}, K^3, E^2, R^2\rangle $ be a circularly
ordered structure, where  $\mathbb Q \times \mathbb Q$ is ordered lexicographically. The symbol  $E$
is defined as follows: $E(a,b)$ holds iff  $a_1=b_1$ for any $a=(a_1, a_2), b=(b_1, b_2)\in M$.
The symbol  $R$ is interpreted as follows:
  $R(a, b)$ holds iff  $a_1=b_1$ and $a_2\le b_2<a_2+\sqrt{2}$ for any $a=(a_1, a_2), b=(b_1, b_2) \in M$.

Obviously, the relation $E(x,y)$ is an equivalence relation
partitioning $M$ into infinitely many infinite convex classes.
Let $f(x):=\rend R(x,M)$. Ob\-vious\-ly, $f(a)\not\in M$ for any $a\in M$,
but $E^*(a,f(a))$. If we consider arbitrary $a,b \in M$ such that $\neg E(a,b)$,
we have both $\neg E^*(a, f(b))$ and $\neg E^*(b,f(a))$.
\end{example}

Let $E(x,y)$ be an equivalence relation partitioning $M$ into infinitely many
infinite convex classes. We say that the induced order on $E$-classes is {\it dense}
if $M/E$ is densely ordered, i.e. for any $a,b\in M$ with $\neg E(a,b)$ there exists $c\in M$
such that $K_0(a,c,b)$, $\neg E(a,c)$ and $\neg E(c,b)$.

In Example \ref{ex_induced} the induced order on $E$-classes is dense.
\vskip 3mm

Let $f$ be a unary function from $M$ to $\overline{M}$.
We say that $f$ is {\it monotonic-to-right (left) on $M$} if it preserves (reverses) the relation
$K_0$, i.e. for any $a,b,c \in M$ such that $K_0(a,b,c)$, we have
$K_0(f(a),f(b),f(c))$ ($K_0(f(c),f(b),f(a))$).



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 is no definable (with parameters)
equivalence relations with infinitely many infinite convex classes.
\vskip 2mm

Let $M$, $N$ be circularly ordered structures.
The {\it $2$-reduct} of $M$ is a circularly ordered
structure with the same universe of $M$ and consisting
of predicates for each $\emptyset$-definable relation on
$M$ 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
$M$ is {\it isomorphic} to $N$ {\it up to
binarity} or {\it binarily isomorphic to} $N$ if the
$2$-reduct of $M$ is isomorphic to the $2$-reduct of
$N$.

\vskip 2mm
We say that a structure $M$ has {\it a trivial definable closure} if $dcl(A)=A$
for any $A\subseteq M$. If $Th(M)$ is binary, the condition of triviality of the
definable closure is just the condition $dcl(\{a\})=\{a\}$ for any $a\in M$.
If additionally $Th(M)$ is 1-transitive, this condition is just
$dcl(\{a\})=\{a\}$ for some $a\in M$.

The following theorem  characterizes up to binarity
$\aleph_0$--categorical 1-tran\-si\-tive non-pri\-mi\-tive weak\-ly cir\-cu\-lar\-ly minimal structures
$M$ of convexity rank greater  than 1 having both a trivial definable closure and
a convex-to-right  formula $R(x,y)$ such that
$r(y):=\rend R(M,y)$ is monotonic-to-right on $M$:

\begin{theorem}\label{th2} {\rm \cite{Kul09}}
Let $M$ be an $\aleph_0$--categorical 1-transitive non-primitive weakly circularly
minimal structure of convexity rank greater than 1,  $dcl(\{a\})$ $=\{a\}$
for some $a\in M$. Sup\-pose that there exists a convex-to-right  formula $R(x,y)$ such that
$r(y):=\rend R(M,y)$ is monotonic-to-right on $M$.
Then  $M$ is isomorphic up to binarity to  $$M'_{s, m, k}:=\langle M,K^3,E_1^2, E^2_2, \ldots, E^2_s, E^2_{s+1}, R^2\rangle,$$
where $M$ is a dense circularly ordered structure,  $s\ge 1$;
$E_{s+1}$ is an equivalence relation partitioning $M$ into $m$ infinite convex classes without endpoints;
$E_i$ for every $1\le i\le s$ is an equivalence relation par\-ti\-tio\-ning 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; $R(M,a)$ has no right endpoint in $M$ and $r^k(a)=a$
for all $a\in M$ and some $k\ge 2$, where   $r^k(y):=r(r^{k-1}(y))$;
for every $1\le i\le s+1$ and any $a\in M$
$$M'_{s,m,k}\models \neg E^*_i(a,r(a))\land \forall y (E_i(y,a)\to \exists u [E^*_i(u,r(a))\land E^*_i(u,r(y))]),$$
$m=1$ or $k$ divides $m$.
\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. In \cite{K_2023}--\cite{KS_2024} algebras of binary isolating formulas
are described for $\aleph_0$-categorical weakly circularly minimal theories of
con\-ve\-xi\-ty rank greater than 1 with a 1-transitive non-primitive automorphism
group and a non-trivial definable closure.
In \cite{SMJ_2024} 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 trivial definable closure.

Recently, in \cite{KMJ_2024} we described algebras of binary isolating formulas
 for $\aleph_0$-categorical 1-transitive
non-primitive weakly circularly minimal structures $M$ with the following restrictions:
\begin{itemize}
\item $RC(x=x)=2$;

\item $dcl(\{a\})=\{a\}$ for some $a\in M$;

\item there exists a monotonic-to-right function to $\overline{M}$;

\item there is no non-trivial equivalence relation partitioning $M$ into finitely many convex classes.
\end{itemize}

 Here we describe algebras of binary isolating formulas
 for these structures replacing the first condition $RC(x=x)=2$ by $RC(x=x)=k$ for some $2\le k<\omega$.



\section{Main Theorem}

\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.
Obviously, strict 1-determinacy of an algebra is equivalent to its determinacy.




\begin{example} \rm Consider the structure $M'_{2,1,2}:=\langle M,K^3,E^2_1, E^2_2,R^2\rangle$ from Theorem \ref{th2}.
Then we have the following: $E_2(x,y)$ is an equivalence relation partitioning $M$ into
infinitely many infinite convex classes so that the induced order on $E_2$-classes is dense;
$E_1(x,y)$ is an equivalence relation partitioning every  $E_2$-class into infinitely many
infinite convex $E_1$-subclasses without endpoints so that the induce order on $E_1$-subclasses is dense;
$R(M,a)$ has no right endpoint in $M$, $r^2(a)=a$ and $\neg E^*(a, r(a))$ for all $a\in M$, and
$r$ preserves $E_i$-classes for every $1\le i\le 2$, i.e. if $E_i(a,b)$ holds for $a,b\in M$ then
both $r(a)$ and $r(b)$ lie in the same $E^*_i$-class.
Whence we obtain that $Th(M'_{2,1,2})$ has eleven binary isolating formulas:
$$\theta_0(x,y):=x=y,$$ $$\theta_1(x,y):=K_0(x,y,r(x))\land E_1(x,y),$$
$$\theta_2(x,y):=K_0(x,y,r(x))\land \neg E_1(x,y) \land E_2(x,y),$$
$$\theta_3(x,y):=K_0(x,y,r(x))\land \neg E_2(x,y) \land \neg E^*_2(y,r(x)),$$
$$\theta_4(x,y):=K_0(x,y,r(x))\land E^*_2(y, r(x)) \land \neg E^*_1(y,r(x)),$$
$$\theta_5(x,y):=K_0(x,y,r(x))\land  E^*_1(y,r(x)),$$
$$\theta_6(x,y):=K_0(r(x),y,x)\land E^*_1(y,r(x)),$$
$$\theta_7(x,y):=K_0(r(x),y,x)\land E^*_2(y,r(x))\land \neg E^*_1(y, r(x)),$$
$$\theta_8(x,y):=K_0(r(x),y,x)\land \neg E_2(x,y) \land \neg E^*_2(y,r(x)),$$
$$\theta_9(x,y):=K_0(r(x),y,x)\land \neg E_2(x,y)\land \neg E_1(x,y),$$
$$\theta_{10}(x,y):=K_0(r(x),y,x)\land E_1(x,y).$$

Obviously, the following holds for any $a\in M$:
$$K_0(\theta_0(a,M),\theta_1(a,M),\theta_2(a,M),\theta_3(a,M), \ldots,\theta_9(a,M), \theta_{10}(a,M)).$$

Define labels for these formulas as follows:
$$\mbox{label } k \mbox{ for } \theta_k(x,y), \mbox{ where } 0\le k\le 10.$$

It easy to check that for the algebra  $\mathfrak{P}_{M'_{2,1,2}}$ the following equalities hold:

$0\cdot k=k\cdot 0=\{k\}$ for every $0\le k\le 10$,

$1\cdot 1=\{1\}$, $1\cdot 2=\{2\}$, $1\cdot 3=\{3\}$, $1\cdot 4=\{4\}$, $1\cdot 5=\{5,6\}$, $1\cdot 6=\{6\}$,

$1\cdot 7=\{7\}$, $1\cdot 8=\{8\}$, $1\cdot 9=\{9\}$, and $1\cdot 10=\{10,0,1\}$.

For example, let's consider the product $1\cdot 5$ in more detail: let $$\phi(x,y):=\exists t [\theta_1(x,t)\land \theta_5(t,y)].$$

Take arbitrary $a,b\in M$ such that  $b\in \theta_1(a,M)$ and consider $\theta_5(b,M)$.
Obviously, $\theta_5(b,M)\cap \theta_5(a,M)\ne \emptyset$ and  $\theta_5(b,M)\cap \theta_6(a,M)\ne \emptyset$,
whence we obtain that  each of $\theta_5(a,y)$ and $\theta_6(a,y)$ is consistent
with  $\phi(a,y)$. Also, we can easily see that each of the remaining $\theta_i(a,y)$ are not
consistent with $\phi(a,y)$. Thus, we have $1\cdot 5=\{5,6\}$.

Further, $2\cdot 1= \{2\}$,  $2\cdot 2= \{2\}$, $2\cdot 3= \{3\}$, $2\cdot 4= \{4,5,6,7\}$, $2\cdot 5= \{7\}$,

$2\cdot 6=\{7\}$, $2\cdot 7=\{7\}$, $2\cdot 8=\{8\}$, $2\cdot 9=\{9,10,0,1,2\}$, and $2\cdot 10=\{2\}$,

$3\cdot 1=\{3\}$,  $3\cdot 2=\{3\}$, $3\cdot 3=\{3,4,5,6,7,8\}$, $3\cdot 4=\{8\}$, $3\cdot 5=\{8\}$,

$3\cdot 6=\{8\}$, $3\cdot 7=\{8\}$, $3\cdot 8=\{8,9,10,0,1,2,3\}$, $3\cdot 9=\{3\}$, and $3\cdot 10=\{3\}$,

$4\cdot 1=\{4\}$, $4\cdot 2=\{4,5,6,7\}$, $4\cdot 3=\{8\}$, $4\cdot 4=\{9\}$, $4\cdot 5=\{9\}$,

$4\cdot 6=\{9\}$, $4\cdot 7=\{9,10,0,1,2\}$, $4\cdot 8=\{3\}$, $4\cdot 9=\{4\}$, and $4\cdot 10=\{4\}$,

$5\cdot 1=\{5,6\}$, $5\cdot 2=\{7\}$, $5\cdot 3=\{8\}$, $5\cdot 4=\{9\}$,  $5\cdot 5=\{10\}$,

$5\cdot 6=\{10,1,2\}$, $5\cdot 7=\{2\}$, $5\cdot 8=\{3\}$, $5\cdot 9=\{5\}$, and $5\cdot 10=\{5\}$,

$6\cdot 1=\{6\}$, $6\cdot 2=\{7\}$, $6\cdot 3=\{8\}$, $6\cdot 4=\{9\}$,  $6\cdot 5=\{10,0,1\}$,

$6\cdot 6=\{1\}$, $6\cdot 7=\{2\}$, $6\cdot 8=\{3\}$, $6\cdot 9=\{4\}$, and $6\cdot 10=\{5,6\}$,

$7\cdot 1=\{7\}$, $7\cdot 2=\{7\}$, $7\cdot 3=\{8\}$, $7\cdot 4=\{9,10,0,1,2\}$,  $7\cdot 5=\{2\}$,

$7\cdot 6=\{2\}$, $7\cdot 7=\{2\}$, $7\cdot 8=\{3\}$, $7\cdot 9=\{4,5,6,7\}$, and $7\cdot 10=\{7\}$,

$8\cdot 1=\{8\}$, $8\cdot 2=\{8\}$, $8\cdot 3=\{8,9,10,0,1,2,3\}$, $8\cdot 4=\{3\}$,  $8\cdot 5=\{3\}$,

$8\cdot 6=\{3\}$, $8\cdot 7=\{3\}$, $8\cdot 8=\{3,4,5,6,7,8\}$, $8\cdot 9=\{8\}$, and $8\cdot 10=\{8\}$,

$9\cdot 1=\{9\}$, $9\cdot 2=\{9,10,0,1,2\}$, $9\cdot 3=\{3\}$, $9\cdot 4=\{4\}$,  $9\cdot 5=\{4\}$,

$9\cdot 6=\{4\}$, $9\cdot 7=\{4,5,6,7\}$, $9\cdot 8=\{8\}$, $9\cdot 9=\{9\}$ and $9\cdot 10=\{9\}$,

$10\cdot 1=\{10,0,1\}$, $10\cdot 2=\{2\}$, $10\cdot 3=\{3\}$, $10\cdot 4=\{4\}$,  $10\cdot 5=\{5\}$,

$10\cdot 6=\{5,6\}$, $10\cdot 7=\{7\}$, $10\cdot 8=\{8\}$, $10\cdot 9=\{9\}$ and $10\cdot 10=\{9\}$.

By these equalities the algebra $\mathfrak{P}_{M'_{2,1,2}}$ is commutative and strictly 7-deterministic.
\end{example}

\begin{theorem}\label{th_s1k}
The algebra $\mathfrak{P}_{M'_{s,1,k}}$ of binary
isolating formulas with mo\-no\-to\-nic-to-right func\-tion $r$ has $2sk+k+1$ labels, is commutative and strictly
$(2s+3)$-deterministic for every $s\ge 1$ and $k\ge 2$.
\end{theorem}

Proof of Theorem \ref{th_s1k}. From Theorem \ref{th2}
$M'_{s,1,k}=\langle M,K^3, E^2_1, \ldots, E^2_s, R^2\rangle$, where $E_s(x,y)$ is an equivalence relation
partitioning $M$ into infinitely many infinite convex classes so that the induced order on $E_s$-classes is dense;
$E_i(x,y)$ for every $1\le i\le s-1$ is an equivalence relation partitioning every $E_{i+1}$-class
into infinitely many convex $E_i$-subclasses without endpoints so that the induced order on $E_i$-subclasses
is dense; $R(M,a)$ has no right endpoint in $M$, $r^k(a)=a$ and $\neg E^*(a, r(a))$ for all $a\in M$;
$r$ preserves $E_i$-classes for every $1\le i\le s$, i.e. if $E_i(a,b)$ holds for $a,b\in M$ then
both $r(a)$ and $r(b)$ lie in the same $E^*_i$-class.
Whence we obtain that the algebra $\mathfrak{P}_{M'_{s,1,k}}$
has $2sk+k+1$ binary isolating formulas:
$$\theta_0(x,y):=x=y,$$
$$\theta_{(2s+1)i+1}(x,y):=K_0(r^i(x),y,r^{i+1}(x))\land E^*_1(y,r^i(x)), \mbox{ where } 0\le i\le k-1,$$
$$\theta_{(2s+1)i+j}(x,y):=K_0(r^i(x),y,r^{i+1}(x))\land E^*_j(y,r^i(x))\land \neg E^*_{j-1}(y, r^i(x)),$$
$$\mbox{ where } 0\le i\le k-1, 2\le j\le s,$$
$$\theta_{(2s+1)i+s+1}(x,y):=K_0(r^i(x),y,r^{i+1}(x))\land \neg E^*_s(y,r^i(x))\land \neg E^*_s(y,r^{i+1}(x)),$$
$$\mbox{ where } 0\le i\le k-1,$$
$$\theta_{(2s+1)(i+1)+1-j}(x,y):=K_0(r^i(x),y,r^{i+1}(x))\land E^*_j(y,r^{i+1}(x))$$
$$\land \neg E^*_{j-1}(y, r^{i+1}(x)), \mbox{ where } 0\le i\le k-1, 2\le j\le s,$$
$$\theta_{(2s+1)(i+1)}(x,y):=K_0(r^i(x),y,r^{i+1}(x))\land E^*_1(y,r^{i+1}(x)),$$ $$\mbox{ where } 0\le i\le k-1.$$

Thus, we have $1+k+(s-1)k +k+(s-1)k+k=2sk+k+1$ binary isolating formulas.
Moreover, we have defined the formulas so that
for any $a\in M$ the following holds:
$$K_0(\theta_0(a,M),\theta_1(a,M),\theta_2(a,M),\ldots, \theta_{2sk+k-1}(a,M),\theta_{2sk+k}(a,M)).$$

We will now prove that the algebra $\mathfrak{P}_{M'_{s,1,k}}$ is commutative and strictly $(2s+3)$-deterministic
 for every $s\ge 1$ and $k\ge 2$.

 First, obviously that $0\cdot l=l\cdot 0=\{l\}$ for any $0\le l\le 3k$.
Suppose further that  $l_1\ne 0$  and $l_2\ne 0$.

Consider the following formula $$\exists t [\theta_{l_1}(x,t)\land \theta_{l_2}(t,y)].$$
\vskip 3mm

{\it Case 1:} $l_1=(2s+1)i_1+1$ for some $0\le i_1\le k-1$.

We have: $K_0(r^{i_1}(x),t,r^{i_1+1}(x))$ and $E^*_1(t, r^{i_1}(x))$.

Let also $l_2=(2s+1)i_2+1$ for some $0\le i_2\le k-1$, i.e.
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$ and $E^*_1(y, r^{i_2}(t))$. Whence we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)) \mbox{ and } E^*_1(y, r^{i_1+i_2}(x)).$$

Clearly, $0\le i_1+i_2\le (k-1)+(k-1)=2k-2$. If $i_1+i_2\le k-1$ then $l_1\cdot l_2=\{(2s+1)(i_1+i_2)+1\}$.
If $i_1+i_2>k-1$ then we have $K_0(r^m(x),y,r^{m+1}(x))$ and $E^*_1(y, r^m(x)),$ where $m=(i_1+i_2)[\rm mod \; k]$.
Then $l_1\cdot l_2=\{(2s+1)m+1\}$.

Let now $l_2=(2s+1)i_2+j$ for some $0\le i_2\le k-1$ and $2\le j\le s$. Then we have:
$K_0(r^{i_2}(t), y,r^{i_2+1}(t))$, $E^*_j(y, r^{i_2}(t))$ and $\neg E^*_{j-1}(y, r^{i_2}(t))$.
Whence we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_j(y, r^{i_1+i_2}(x)) \mbox{ and } \neg E^*_{j-1}(y, r^{i_1+i_2}(x)).$$

If $i_1+i_2\le k-1$ then $l_1\cdot l_2=\{(2s+1)(i_1+i_2)+j\}$.
If $i_1+i_2>k-1$ then we have $K_0(r^m(x),y,r^{m+1}(x))$, $E^*_j(y, r^m(x))$ and $\neg E^*_{j-1}(y, r^m(x)),$
where $m=(i_1+i_2)[\rm mod \; k]$.
Then $l_1\cdot l_2=\{(2s+1)m+j\}$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_j(t, r^{i_2}(x))$, $\neg E^*_{j-1}(t, r^{i_2}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$ and $E^*_1(y, r^{i_1}(t))$.

Whence we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_j(y, r^{i_1+i_2}(x)) \mbox{ and } \neg E^*_{j-1}(y, r^{i_1+i_2}(x)).$$

Then $l_2\cdot l_1=\{(2s+1)m+j\}$, where $m=(i_1+i_2)[\rm mod \; k]$.

Let now $l_2=(2s+1)i_2+s+1$ for some $0\le i_2\le k-1$, i.e.
$K_0(r^{i_2}(t)$, $y$, $r^{i_2+1}(t))$, $\neg E^*_s(y, r^{i_2}(t))$ and $\neg E^*_s(y, r^{i_2+1}(t))$.
Whence we obtain:
$$K_0(r^{i_1+i_2}(x), y,r^{i_1+i_2+1}(x)), \neg E^*_s(y, r^{i_1+i_2}(x)) \mbox{ and } \neg E^*_s(y, r^{i_1+i_2+1}(x)).$$

Let $m=(i_1+i_2)[\rm mod \; k]$. then $l_1\cdot l_2=\{(2s+1)m+s+1\}$.

Consider the product $l_2\cdot l_1$. We have:
$K_0(r^{i_2}(x)$, $t$, $r^{i_2+1}(x))$,\\ $\neg E^*_s(t, r^{i_2}(x))$, $\neg E^*_s(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$ and $E^*_1(y, r^{i_1}(t))$.

Whence we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_s(y, r^{i_1+i_2}(x)) \mbox{ and } \neg E^*_s(y, r^{i_1+i_2}(x)).$$

Then $l_2\cdot l_1=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2)[\rm mod \; k]$.

Let now $l_2=(2s+1)(i_2+1)+1-j$ for some $0\le i_2\le k-1$ and $2\le j\le s$, i.e.
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$, $E^*_j(y, r^{i_2+1}(t))$ and $\neg E^*_{j-1}(y, r^{i_2+1}(t))$.
Whence we obtain:
$$K_0(r^{i_1+i_2}(t),y,r^{i_1+i_2+1}(t)), E^*_j(y, r^{i_1+i_2+1}(t)) \mbox{ and } \neg E^*_{j-1}(y, r^{i_1+i_2+1}(t)).$$

Let $m=(i_1+i_2)[\rm mod \; k]$. Then $l_1\cdot l_2=\{(2s+1)(m+1)+1-j\}$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_j(t, r^{i_2}(x))$, $\neg E^*_{j-1}(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$ and $E^*_1(y, r^{i_1}(t))$.

Whence we obtain:
$$K_0(r^{i_1+i_2}(t),y,r^{i_1+i_2+1}(t)), E^*_j(y, r^{i_1+i_2+1}(t)) \mbox{ and } \neg E^*_{j-1}(y, r^{i_1+i_2+1}(t)).$$

Then $l_2\cdot l_1=\{(2s+1)(m+1)+1-j\}$, where $m=(i_1+i_2)[\rm mod \; k]$.

Let now $l_2=(2s+1)(i_2+1)$ for some $0\le i_2\le k-1$, i.e.
$K_0(r^{i_2}(t)$, $y$, $r^{i_2+1}(t))$ and $E^*_1(y, r^{i_2+1}(t))$. Whence we obtain:
$E^*_1(y,r^{i_1+i_2+1}(x))$  and either $K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x))$  or
$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x))$.

Let $m=(i_1+i_2+1)[\rm mod \; k]$. Clearly, $0\le m\le k-1$, since $1\le i_1+i_2+1\le (k-1)+(k-1)+1=2k-1$.
If $m=0$ then $l_1\cdot l_2=\{0,1,2sk+k\}$. If $m\ne 0$ then $l_1\cdot l_2=\{(2s+1)(m+1), (2s+1)(m+1)+1\}$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_1(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$ and $E^*_1(y, r^{i_1}(t))$. Whence we obtain:
$E^*_1(y,r^{i_1+i_2+1}(x))$  and either $K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x))$  or
$K_0(r^{i_1+i_2+1}(x)$, $y$, $r^{i_1+i_2+2}(x))$.

Let $m=(i_1+i_2+1)[\rm mod \; k]$. Similarly, we have: if $m=0$,
$l_2\cdot l_1=\{0,1,2sk+k\}$. If $m\ne 0$, $l_2\cdot l_1=\{(2s+1)(m+1), (2s+1)(m+1)+1\}$.
\vskip 3mm

{\it Case 2.} $l_1=(2s+1)i_1+j_1$ for some $0\le i_1\le k-1$ and $2\le j_1\le s$.

We have the following: $K_0(r^{i_1}(x),t,r^{i_1+1}(x))$, $E^*_{j_1}(t, r^{i_1}(x))$ and
$\neg E^*_{j_1-1}(t$, $r^{i_1}(x))$.

Let also $l_2=(2s+1)i_2+j_2$ for some $0\le i_2\le k-1$ and $2\le j_2\le s$. Then we have
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$, $E^*_{j_2}(y, r^{i_2}(t))$ and
$\neg E^*_{j_2-1}(y, r^{i_2}(t))$.

Let $m=(i_1+i_2)[\rm mod \; k]$. If $j_1\le j_2$, we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_{j_2}(y, r^{i_1+i_2}(x)) \mbox{ and }
\neg E^*_{j_2-1}(y, r^{i_1+i_2}(x)),$$
whence $l_1\cdot l_2=\{(2s+1)m+j_2\}$. If $j_1>j_2$, we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_{j_1}(y, r^{i_1+i_2}(x)) \mbox{ and }
\neg E^*_{j_1-1}(y, r^{i_1+i_2}(x)),$$ whence $l_1\cdot l_2=\{(2s+1)m+j_1\}$.

Let now $l_2=(2s+1)i_2+s+1$ for some $0\le i_2\le k-1$. Then we have
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$, $\neg E^*_s(y, r^{i_2}(t))$ and
$\neg E^*_s(y, r^{i_2+1}(t))$. Whence we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), \neg E^*_s(y, r^{i_1+i_2}(x)) \mbox{ and }
\neg E^*_s(y, r^{i_1+i_2+1}(x)).$$
Then $l_1\cdot l_2=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2)[\rm mod\; k]$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $\neg E^*_s(t, r^{i_2}(x))$, $\neg E^*_s(t, r^{i_2+1}(x))$,
$\;K_0(r^{i_1}(t),y,r^{i_1+1}(t))$, $\;E^*_{j_1}(y, r^{i_1}(t))$ and
$\neg E^*_{j_1-1}(y, r^{i_1}(t))$. Whence we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), \neg E^*_s(y, r^{i_1+i_2}(x)) \mbox{ and }
\neg E^*_s(y, r^{i_1+i_2+1}(x)).$$
Similarly, $l_2\cdot l_1=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2)[\rm mod\; k]$.

Let now $l_2=(2s+1)(i_2=1)+1-j_2$ for some $0\le i_2\le k-1$ and $2\le j_2\le s$.
We have the following: $K_0(r^{i_2}(t),y,r^{i_2+1}(t))$, $E^*_{j_2}(y, r^{i_2+1}(t))$ and
$\neg E^*_{j_2-1}(y, r^{i_2+1}(t))$.

If $j_1\le j_2$, we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_{j_2}(y, r^{i_1+i_2+1}(x)) \mbox{ and }
\neg E^*_{j_2-1}(y, r^{i_1+i_2+1}(x)).$$
Then $l_1\cdot l_2=\{(2s+1)m+1-j_2\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.
If $j_1>j_2$, we obtain:
$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+1}(x))$  and
$\neg E^*_{j_1-1}(y, r^{i_1+i_2+1}(x))$, whence $l_1\cdot l_2=\{(2s+1)m+j_1\}$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_{j_2}(t, r^{i_2+1}(x))$, $\neg E^*_{j_2-1}(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t)), E^*_{j_1}(y, r^{i_1}(t))$ and
$\neg E^*_{j_1-1}(y, r^{i_1}(t))$.

Let $m=(i_1+i_2+1)[\rm mod\; k]$. If $j_1\le j_2$, we obtain:
$$K_0(r^{i_1+i_2}(x),y,r^{i_1+i_2+1}(x)), E^*_{j_2}(y, r^{i_1+i_2+1}(x)) \mbox{ and }
\neg E^*_{j_2-1}(y, r^{i_1+i_2+1}(x)),$$
whence $l_2\cdot l_1=\{(2s+1)m+1-j_2\}$. If $j_1>j_2$, we obtain:
$K_0(r^{i_1+i_2+1}(x)$, $y$, $r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+1}(x))$  and
$\neg E^*_{j_1-1}(y, r^{i_1+i_2+1}(x))$, whence $l_2\cdot l_1=\{(2s+1)m+j_1\}$.

Let now $l_2=(2s+1)(i_2+1)$ for some $0\le i_2\le k-1$. Then we have:
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$ and $E^*_1(y, r^{i_2+1}(t))$. Whence we obtain:
$K_0(r^{i_1+i_2+1}(x)$, $y,r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+1}(x))$ and
$\neg E^*_{j_1-1}(y, r^{i_1+i_2+1}(x))$.
Then $l_1\cdot l_2=\{(2s+1)m+j_1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_1(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$, $E^*_{j_1}(y, r^{i_1}(t))$  and
$\neg E^*_{j_1-1}(y, r^{i_1}(t))$.

Whence we obtain:
$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+1}(x))$  and
$\neg E^*_{j_1-1}(y, r^{i_1+i_2+1}(x))$,
whence $l_2\cdot l_1=\{(2s+1)m+j_1\}$.
\vskip 3mm

{\it Case 3.} $l_1=(2s+1)i_1+s+1$ for some $0\le i_1\le k-1$.

We have the following:  $K_0(r^{i_1}(x),t,r^{i_1+1}(x))$, $\neg E^*_s(t, r^{i_1}(x))$  and
$\neg E^*_s(t$, $r^{i_1+1}(x))$.

Let also $l_2=(2s+1)i_2+s+1$ for some $0\le i_2\le k-1$, i.e.
$K_0(r^{i_2}(t)$, $y$, $r^{i_2+1}(t))$, $\neg E^*_s(y, r^{i_2}(t))$  and
$\neg E^*_s(y, r^{i_2+1}(t))$.

Whence we obtain:
$K_0(r^{i_1+i_2}(x),t,r^{i_1+i_2+2}(x))$.

Let $m=(i_1+i_2+1)[\rm mod\; k]$. If $m=0$ then
$$l_1\cdot l_2=\{(2s+1)k-s, (2s+1)k-s+1, \ldots,  (2s+1)k, 0,1, \ldots, s, s+1\},$$
i.e.  the product $l_1\cdot l_2$ consists of $2s+3$ labels. If $m\ne 0$ then
$$l_1\cdot l_2=\{(2s+1)m+s+1, (2s+1)m+s+2, \ldots, (2s+1)m+3s+2\},$$
i.e.  the product $l_1\cdot l_2$ consists of $2s+1$ labels.

Let now $l_2=(2s+1)(i_2=1)+1-j$ for some $0\le i_2\le k-1$ and $2\le j\le s$. We have the following:
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$, $E^*_j(y, r^{i_2+1}(t))$ and $\neg E^*_{j-1}(y, r^{i_2+1}(t))$.
Whence we obtain:
$$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x)), \neg E^*_s(y, r^{i_1+i_2+1}(t)) \mbox{ and }
\neg E^*_s(y, r^{i_1+i_2+2}(t)).$$
Then $l_1\cdot l_2=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_j(t, r^{i_2+1}(x))$, $\neg E^*_{j-1}(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$, $\neg E^*_s(y, r^{i_1}(t))$ and $\neg E^*_s(y, r^{i_1+1}(t))$.
Whence we obtain:
$$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x)), \neg E^*_s(y, r^{i_1+i_2+1}(t)) \mbox{ and }
\neg E^*_s(y, r^{i_1+i_2+2}(t)).$$
Then also $l_2\cdot l_1=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.

Let now $l_2=(2s+1)(i_2+1)$ for some $0\le i_2\le k-1$. Then we have:
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$ and $E^*_1(y, r^{i_2+1}(t))$. Whence we obtain:
$$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x)), \neg E^*_s(y, r^{i_1+i_2+1}(t)) \mbox{ and }
\neg E^*_s(y, r^{i_1+i_2+2}(t)).$$
Then $l_1\cdot l_2=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.

Consider the product $l_2\cdot l_1$. We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_1(t, r^{i_2+1}(x))$,
$K_0(r^{i_1}(t),y,r^{i_1+1}(t))$, $\neg E^*_s(y, r^{i_1}(t))$ and $\neg E^*_s(y, r^{i_1+1}(t))$.
Whence we obtain:
$$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x)), \neg E^*_s(y, r^{i_1+i_2+1}(t)) \mbox{ and }
\neg E^*_s(y, r^{i_1+i_2+2}(t)).$$
Then also $l_2\cdot l_1=\{(2s+1)m+s+1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.
\vskip 3mm

{\it Case 4.} $l_1=(2s+1)(i_1+1)+1-j_1$ for some $0\le i_1\le k-1$ and $2\le j_1\le s$.

We have the following: $$K_0(r^{i_1}(x),t,r^{i_1+1}(x)), E^*_{j_1}(t, r^{i_1+1}(x))
\mbox{ and } \neg E^*_{j_1-1}(t, r^{i_1+1}(x)).$$

Let also $l_2=(2s+1)(i_2+1)+1-j_2$ for some $0\le i_2\le k-1$ and $2\le j_2\le s$, i.e.
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$, $E^*_{j_2}(y, r^{i_2+1}(t))$
and $\neg E^*_{j_2-1}(y, r^{i_2+1}(t))$.

Let $m=(i_1+i_2+1)[\rm mod\; k]$.

If $j_1\le j_2$, we obtain:
$K_0(r^{i_1+i_2+1}(x)$, $y$, $r^{i_1+i_2+2}(x))$, $E^*_{j_2}(y, r^{i_1+i_2+2}(x))$
 and  $\neg E^*_{j_2-1}(y, r^{i_1+i_2+2}(x))$,
whence $l_1\cdot l_2=(2s+1)(m+1)+1-j_2$.

 If $j_1>j_2$, we obtain:
$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+2}(x))$
 and $\neg E^*_{j_1-1}(y, r^{i_1+i_2+2}(x))$,
whence $l_1\cdot l_2=(2s+1)(m+1)+1-j_1$.

Let now $l_2=(2s+1)(i_2+1)$ for some $0\le i_2\le k-1$. Then we have:
$K_0(r^{i_2}(t)$, $y$, $r^{i_1+1}(t))$ and $E^*_1(y, r^{i_2+1}(t))$.

Whence we obtain:
$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+2}(x)$
 and  $\neg E^*_{j_1-1}(y, r^{i_1+i_2+2}(x))$.
Then $l_1\cdot l_2=\{(2s+1)(m+1)+1-j_1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.

Consider the product $l_2\cdot l_1$.

We have the following:
$K_0(r^{i_2}(x),t,r^{i_2+1}(x))$, $E^*_1(t, r^{i_2+1}(x))$,
$\;K_0(r^{i_1}(t)$, $y$, $r^{i_1+1}(t))$, $\;E^*_{j_1}(y, r^{i_1+1}(t))$ and $\;\neg E^*_{j_1-1}(y$, $r^{i_1+1}(t))$.

Whence we obtain:
$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x))$, $E^*_{j_1}(y, r^{i_1+i_2+1}(x))$  and
$\neg E^*_{j_1-1}(y, r^{i_1+i_2+2}(x))$.

Then also $l_2\cdot l_1=\{(2s+1)(m+1)+1-j_1\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.
\vskip 3mm

{\it Case 5.} $l_1=(2s+1)(i_1+1)$ for some $0\le i_1\le k-1$.

We have the following: $K_0(r^{i_1}(x),t,r^{i_1+1}(x))$ and $E^*_1(t, r^{i_1+1}(x))$.

Let also $l_2=(2s+1)(i_2+1)$ for some $0\le i_2\le k-1$. Then we have:
$K_0(r^{i_2}(t),y,r^{i_2+1}(t))$ and $E^*_1(y, r^{i_2+1}(t))$. Whence we obtain:
$$K_0(r^{i_1+i_2+1}(x),y,r^{i_1+i_2+2}(x)) \mbox{ and } E^*_1(y, r^{i_1+i_2+2}(x)).$$
Then $l_1\cdot l_2=\{(2s+1)(m+1)\}$, where $m=(i_1+i_2+1)[\rm mod\; k]$.

Thus, we established that the algebra $\mathfrak{P}_{M'_{s,1,k}}$ is commutative and strictly $(2s+3)$-deterministic
 for every $s\ge 1$ and $k\ge 2$.
\endproof

\medskip
In conclusion, the authors thank the anonymous reviewer for useful
comments and constructive criticism that contributed to improving
the presentation.

\bigskip
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