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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
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\title[Almost $n$-ary and almost $n$-aritizable theories]{Almost $n$-ary and almost $n$-aritizable theories}
\author{S.V. Sudoplatov}%



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


\thanks{\sc Sudoplatov, S.V.,
Almost $n$-ary and almost $n$-aritizable theories}
\thanks{\copyright \ 2022 Sudoplatov S.V}
\thanks{\rm The work of the author was carried out in the framework of the
State Contract of the Sobolev Institute of Mathematics, Project
No.~FWNF-2022-0012.}
\thanks{\it Received  December, ??, 2021, published  ???, ??,  2022.}%


\semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} We study possibilities for almost $n$-ary
and $n$-aritizable theories. Their dynamics both in general case,
for $\omega$-categorical theories, and with respect to operations
for theories are described.\medskip

\noindent{\bf Keywords:} elementary theory, almost $n$-ary theory,
almost $n$-aritizable theory.
 \end{quote}
}

\bigskip
We continue to study arities of theories and of their expansions
\cite{aafot}. In the present paper we introduce natural notions of
almost $n$-ary and almost $n$-aritizable theories, and describe
their dynamics both in general case, for $\omega$-categorical
theories, and with respect to operations for theories.

\section{Preliminaries}

Recall a series of notions related to arities and aritizabilities
of theories.

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

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

\medskip
{\bf Definition} \cite{ZPS, CCMCT}. {\rm Let $\Delta$ be a set of
formulae of a theory $T$, and $p(\bar{x})$ a type of $T$ lying in
$S(T)$. The type $p(\bar{x})$ is said to be
\emph{$\Delta$-based}\index{Type!$\Delta$-based} if $p(\bar{x})$
is isolated by a set of formulas $\varphi^\delta\in p$, where
$\varphi\in\Delta$, $\delta\in\{0,1\}$.}

\medskip
The following lemma, being a corollary of Compactness Theorem,
noticed in \cite{ZPS}.

\medskip
%\begin{lemma}\label{le123_126}
\begin{lemma}\label{lem01} A  theory  $T$  is
$\Delta$-based  if and only if,  for  any  tuple~$\bar{a}$ of any
{\rm (}some{\rm )} weakly saturated model of $T$, the type ${\rm
tp}(\bar{a})$ is $\Delta$-based.
\end{lemma}

\medskip
{\bf Definition} \cite{aafot}. An elementary theory $T$ is called
{\em unary}, or {\em $1$-ary}, if any $T$-formula
$\varphi(\overline{x})$ is $T$-equivalent to a Boolean combination
of $T$-formulas, each of which is of one free variable, and of
formulas of form $x\approx y$.

For a natural number $n\geq 1$, a formula $\varphi(\overline{x})$
of a theory $T$ is called {\em $n$-ary}, or an {\em $n$-formula},
if $\varphi(\overline{x})$ is $T$-equivalent to a Boolean
combination of $T$-formulas, each of which is of $n$ free
variables.

For a natural number $n\geq 2$, an elementary theory $T$ is called
{\em $n$-ary}, or an {\em $n$-theory}, if any $T$-formula
$\varphi(\overline{x})$ is $n$-ary.

A theory $T$ is called {\em binary} if $T$ is $2$-ary, it is
called {\em ternary} if $T$ is $3$-ary, etc.

We will admit the case $n=0$ for $n$-formulae
$\varphi(\overline{x})$. In such a case $\varphi(\overline{x})$ is
just $T$-equivalent to a sentence
$\forall\overline{x}\varphi(\overline{x})$.

If $T$ is a theory such that $T$ is $n$-ary and not $(n-1)$-ary
then the value $n$ is called the arity of $T$ and it is denoted by
${\rm ar}(T)$. If $T$ does not have any arity we put ${\rm
ar}(T)=\infty$.

Similarly, for a formula $\varphi$ of a theory $T$ we denote by
${\rm ar}_T(\varphi)$ the natural value $n$ if $\varphi$ is
$n$-ary and not $(n-1)$-ary. If a theory $T$ is fixed we write
${\rm ar}(\varphi)$ instead of ${\rm ar}_T(\varphi)$.

\medskip
Clearly, ${\rm ar}(\varphi)\leq|{\rm FV}(\varphi)|$, where ${\rm
FV}(\varphi)$ is the set of free variables of formula $\varphi.$

\medskip
The following example illustrates the notions above, and it will
be used below.

\begin{example}\label{excircular} {\rm Recall \cite{KulMac, Kulp3, Kulp4}
that a {\em circular}, or {\em cyclic}  order relation is
described by a ternary relation $K_3$ satisfying the following conditions:\\
 (co1) $\forall x\forall y \forall z (K_3(x,y,z)\to K_3(y,z,x));$\\
 (co2) $\forall x\forall y \forall z (K_3(x,y,z)\land K_3(y,x,z)
 \leftrightarrow x=y \lor y=z \lor z=x);$\\
 (co3) $\forall x\forall y \forall z(K_3(x,y,z)\to \forall t[K_3(x,y,t)
 \lor K_3(t,y,z)]);$\\
 (co4) $\forall x\forall y \forall z (K_3(x,y,z)\lor K_3(y,x,z)).$

Clearly, ${\rm ar}(K_3(x,y,z))=3$ if the relation has at least
three element domain. Hence, theories with infinite circular order
relations are at least $3$-ary.

The following generalization of circular order produces a {\em
$n$-ball}, or {\em $n$-spherical}, or {\em $n$-circular} order
relation, for $n\geq 4$, which is described by a $n$-ary relation
$K_n$ satisfying the following conditions:\\
 (nbo1) $\forall x_1,\ldots,x_n (K_n(x_1,x_2,\ldots,x_n)\to K_n(x_2,\ldots,x_n,x_1));$\\
 (nbo2) $\forall x_1,\ldots,x_n \biggm(K_n(x_1,\ldots,x_i,x_{i+1},\ldots,x_n)\land$  $$\land K_n(x_1,\ldots,x_{i+1},x_{i},\ldots,x_n)
 \leftrightarrow\bigvee\limits_{i=1}^{n-1} x_i=x_{i+1}\biggm);$$\\
 (nbo3) $\forall x_1,\ldots,x_n(K_n(x_1,\ldots,x_n)\to \forall t[K_n(x_1,\ldots,x_{n-1},t)
 \lor K_n(t,x_2,\ldots,x_n)]);$\\
 (nbo4) $\forall x_1,\ldots,x_n (K_n(x_1,\ldots,x_i,x_{i+1},\ldots,x_n)\lor K_n(x_1,\ldots,x_{i+1},x_{i},\ldots,x_n)),$ $i<n.$

Clearly, ${\rm ar}(K_n(x_1,\ldots,x_n))=n$ if the relation has at
least $n$-element domain. Thus, theories with infinite $n$-ball
order relations are at least $n$-ary.}
\end{example}

{\bf Definition} \cite{aafot}. A $T$-formula
$\varphi(\overline{x})$ is called {\em $n$-expansible}, or {\em
$n$-arizable}, or {\em $n$-aritizable}, if $T$ has an expansion
$T'$ such that $\varphi(\overline{x})$ is $T'$-equivalent to a
Boolean combination of $T'$-formulas with $n$ free variables.

A theory $T$ is called {\em $n$-expansible}, or {\em
$n$-arizable}, or {\em $n$-aritizable}, if there is an $n$-ary
expansion $T'$ of $T$.

A theory $T$ is called {\em arizable} or {\em aritizable}, if $T$
is $n$-aritizable for some $n$.

A $1$-aritizable theory is called {\em unary-able}, or {\em
unary-tizable}. A $2$-aritizable theory is called {\em
binary-tizable} or {\em binarizable}, a $3$-aritizable theory is
called {\em ternary-tizable} or {\em ternarizable}, etc.

\medskip
{\bf Definition.} \cite{Wo} The {\em disjoint
union}\index{Disjoint union!of structures}
$\bigsqcup\limits_{n\in\omega}\mathcal{
M}_n$\index{$\bigsqcup\limits_{n\in\omega}\mathcal{ M}_n$} of
pairwise disjoint structures $\mathcal{ M}_n$ for pairwise
disjoint predicate languages $\Sigma_n$, $n\in\omega$, is the
structure of language
$\bigcup\limits_{n\in\omega}\Sigma_n\cup\{P^{(1)}_n\mid
n\in\omega\}$ with the universe $\bigsqcup\limits_{n\in\omega}
M_n$, $P_n=M_n$, and interpretations of predicate symbols in
$\Sigma_n$ coinciding with their interpretations in $\mathcal{
M}_n$, $n\in\omega$. The {\em disjoint union of
theories}\index{Disjoint union!of theories} $T_n$ for pairwise
disjoint languages $\Sigma_n$ accordingly, $n\in\omega$, is the
theory
$$\bigsqcup\limits_{n\in\omega}T_n\rightleftharpoons{\rm Th}\left(\bigsqcup\limits_{n\in\omega}\mathcal{ M}_n\right),$$
where\index{$\bigsqcup\limits_{n\in\omega}T_n$} $\mathcal{
M}_n\models T_n$, $n\in\omega$. Taking empty sets instead of some
structures $\mathcal{ M}_k$ we obtain disjoint unions of finitely
many structures and theories. In particular, we have the disjoint
unions $\mathcal{ M}_0\sqcup\ldots\sqcup\mathcal{ M}_n$ and their
theories $T_0\sqcup\ldots\sqcup T_n$.

\begin{theorem}\label{thdu} {\rm \cite{aafot}.}
$1.$ For any theories $T_m$, $m\in\omega$, and their disjoint
union $\bigsqcup\limits_{m\in\omega}T_m$, all $T_m$ are
$n$-theories iff $\bigsqcup\limits_{m\in\omega}T_m$ is an
$n$-theory, moreover, $${\rm
ar}\left(\bigsqcup\limits_{m\in\omega}T_m\right)={\rm max}\{{\rm
ar}(T_m)\mid m\in\omega\}.$$

$2.$ For any theories $T_m$, $m\in\omega$, and their disjoint
union $\bigsqcup\limits_{m\in\omega}T_m$, all $T_m$ are
$n$-aritizable iff $\bigsqcup\limits_{m\in\omega}T_m$ is
$n$-aritizable.
\end{theorem}

{\bf Definition} \cite{EKS20}. Let $\mathcal{M}$ and $\mathcal{N}$
be structures of relational languages $\Sigma_\mathcal{M}$ and
$\Sigma_\mathcal{N}$ respectively. We define the {\em composition}
$\mathcal{M}[\mathcal{N}]$ of $\mathcal{M}$ and $\mathcal{N}$
satisfying the following conditions:

1)
$\Sigma_{\mathcal{M}[\mathcal{N}]}=\Sigma_\mathcal{M}\cup\Sigma_\mathcal{N}$;

2) $M[N]=M\times N$, where $M[N]$, $M$, $N$ are universes of
$\mathcal{M}[\mathcal{N}]$, $\mathcal{M}$, and $\mathcal{N}$
respectively;

3) if $R\in\Sigma_\mathcal{M}\setminus\Sigma_\mathcal{N}$,
$\mu(R)=n$, then $((a_1,b_1),\ldots,(a_n,b_n))\in
R_{\mathcal{M}[\mathcal{N}]}$ if and only if $(a_1,\ldots,a_n)\in
R_{\mathcal{M}}$;

4) if $R\in\Sigma_\mathcal{N}\setminus\Sigma_\mathcal{M}$,
$\mu(R)=n$, then $((a_1,b_1),\ldots,(a_n,b_n))\in
R_{\mathcal{M}[\mathcal{N}]}$ if and only if $a_1=\ldots =a_n$ and
$(b_1,\ldots,b_n)\in R_{\mathcal{N}}$;

5) if $R\in\Sigma_\mathcal{M}\cap\Sigma_\mathcal{N}$, $\mu(R)=n$,
then $((a_1,b_1),\ldots,(a_n,b_n))\in
R_{\mathcal{M}[\mathcal{N}]}$ if and only if $(a_1,\ldots,a_n)\in
R_{\mathcal{M}}$, or $a_1=\ldots =a_n$ and $(b_1,\ldots,b_n)\in
R_{\mathcal{N}}$.

The composition $\mathcal{M}[\mathcal{N}]$ is called {\em
$e$-definable}, or {\em {\rm equ}-definable}, if
$\mathcal{M}[\mathcal{N}]$ has an $\emptyset$-definable
equivalence relation $E$ whose $E$-classes are universes of the
copies of $\mathcal{N}$ forming $\mathcal{M}[\mathcal{N}]$. If the
equivalence relation $E$ is fixed, the $e$-definable composition
is called {\em $E$-definable}.

\medskip
Using a nice basedness of $E$-definable compositions $T_1[T_2]$
(see \cite{EKS20}) till the formulas of form $E(x,y)$ and
generating formulas for $T_1$ and $T_2$ we have the following:

\begin{theorem}\label{thcomp} {\rm \cite{aafot}.} $1.$  For any theories $T_1$ and $T_2$ and
their $E$-definable composition $T_1[T_2]$, $T_1$ and $T_2$ are
$n$-theories, for $n\geq 2$, iff $T_1[T_2]$ is an $n$-theory,
moreover, ${\rm ar}(T_1[T_2])={\rm max}\{{\rm ar}(T_1),{\rm
ar}(T_2)\}$, if models of $T_1$ and of $T_2$ have at least two
elements, and ${\rm ar}(T_1[T_2])={\rm max}\{{\rm ar}(T_1),{\rm
ar}(T_2),2\}$, if a model of $T_1$ or $T_2$ is a singleton.

$2.$ For any theories $T_1$ and $T_2$ and their $E$-definable
composition $T_1[T_2]$, $T_1$ and $T_2$ are $n$-aritizable iff
$T_1[T_2]$ is $n$-aritizable.
\end{theorem}


\section{Almost $n$-ary and $n$-aritizable theories, their dynamics}

{\bf Definition.} (Cf. \cite{Kulp3, Kulp4}) A theory $T$ is called
{\em almost $n$-ary} if there are finitely many formulae
$\varphi_1(\overline{x}),\ldots,\varphi_m(\overline{x})$ such that
each $T$-formula is $T$-equivalent to a Boolean combination of
$n$-formulae and formulae obtained by substitutions of free
variables in
$\varphi_1(\overline{x}),\ldots,\varphi_m(\overline{x})$.

In such a case we say that the formulae
$\varphi_1(\overline{x}),\ldots,\varphi_m(\overline{x})$ witness
that $T$ is almost $n$-ary.

Almost $1$-ary theories are called {\em almost unary}, almost
$2$-ary theories are called {\em almost binary}, almost $3$-ary
theories are called {\em almost ternary}, etc.

A theory $T$ is called {\em almost $n$-aritizable} if some
expansion $T'$ of $T$ is almost $n$-ary.

Almost $1$-aritizable theories are called {\em almost
unary-tizable}, almost $2$-aritizable theories are called {\em
almost binarizable}, almost $3$-aritizable theories are called
{\em almost ternarizable}, etc.

\medskip
The following properties are obvious.

\medskip 1. Any $n$-ary (respectively, $n$-aritizable) theory is
almost $n$-ary (almost $n$-ari\-ti\-z\-a\-ble).

\medskip
2. Any almost $n$-ary (respectively, $n$-aritizable) theory is
almost $k$-ary (almost $k$-aritizable) for any $k\geq n$.

\medskip
3. Any theory of a finite structure is almost unary.

\medskip
Families of weakly circularly minimal structures produce examples
of almost binary theories which are not binary \cite{Kulp3,
Kulp4}. Similarly natural generalizations of weakly circularly
minimal structures till $n$-circular orders give examples of
almost $(n-1)$-ary theories $T_n$ with ${\rm ar}(T_n)=n$, $n\geq
4$.

\medskip
Assuming that the witnessing set
$\{\varphi_1(\overline{x}),\ldots,\varphi_m(\overline{x})\}$ is
minimal for the almost $n$-ary theory $T$ we have either $m=0$ of
$l(\overline{x})>n$.

Thus we have two minimal characteristics witnessing the almost
$n$-arity of $T$: $m$ and $l(\overline{x})$. The pair
$(m,l(\overline{x}))$ is called the {\em degree} of the almost
$n$-arity of $T$, or the {\em {\rm aar}-degree} of $T$, denoted by
${\rm deg}_{\rm aar}(T)$. Here we assume that $n$ is minimal with
almost $n$-arity of $T$, this $n$ is denoted by ${\rm aar}(T)$.
Clearly, ${\rm aar}(T)\leq{\rm ar}(T)$, and if $m=0$, i.e.,
$n={\rm ar}(T)={\rm aar}(T)$ then it is supposed that
$l(\overline{x})=0$, too.

We have ${\rm aar}(T)\in\omega$ if and only if ${\rm
ar}(T)\in\omega$. So if ${\rm ar}(T)=\infty$ then it is natural to
put ${\rm aar}(T)=\infty$.

Besides, $n={\rm ar}(T)={\rm aar}(T)\in\omega$ means that the set
$\Delta_n(T)$ of $T$-formulae with $n$ free variables allows to
express all $\emptyset$-definable sets for $T$ by Boolean
combinations and taking any set $\Delta_k(T)$ of $T$-formulae with
$k<n$ free variables all $\emptyset$-definable sets for $T$ can be
expressed by Boolean combinations of formulae in $\Delta_k$ and
substitutions of infinitely many formulae $\varphi(\overline{x})$
only, where $l(\overline{x})=n$.

By the definition if ${\rm aar}(T)=n$ then
\begin{equation}\label{eq_aar}{\rm
deg}_{\rm ar}(T)\in\{(0,0)\}\cup\{(m,r)\mid
m\in\omega\setminus\{0\}, r\in\omega, r>n\}.
\end{equation}
The described pairs in the relation (\ref{eq_aar}) are called {\em
admissible}.

\begin{theorem}\label{th_aar_mn}
For any $m,n\in\omega\setminus\{0\}$ with $m\leq n$ there is a
theory $T_{mn}$ with ${\rm aar}(T{mn})=m$ and ${\rm
ar}(T_{mn})=n$.
\end{theorem}

Proof. We use disjoint unions of {\em dense} $n$-spherically
ordered theories $T_n$ with infinite orders, i.e., theories,
generated by $n$-spherical orders $K_n(x_1,x_2,x_3,\ldots,x_n)$
satisfying the axioms $$\forall x_1,x_2,\ldots
x_n\Bigg(K_n(x_1,x_2,x_3,\ldots,x_n)\wedge\bigwedge_{i\ne j}\neg
x_i\approx x_j\to$$
$$\to\exists y\Bigg(\bigwedge_{i\leq n}\neg
x_i\approx y\wedge K_n(x_1,y,x_3,\ldots,x_n)\Bigg)\Bigg).$$ For
$n=2$ we take the theory $T_2$ of dense linear order
$K_2(x_1,x_2)$ without endpoints, having ${\rm ar}(T_2)=2$, and
for $n=1$~--- the theory $T_1$ of the empty languages, having
${\rm ar}(T_1)=1$.

Similarly to dense linear orders and dense circular orders, dense
$n$-spherical orders produce quantifier eliminations with ${\rm
ar}(T_n)=n$, $n\in\omega\setminus\{0\}$.

Using Theorem \ref{thdu} we obtain $m={\rm ar}(T)={\rm aar}(T)$
taking $T_{mm}=\bigsqcup\limits_{r\in\omega}T^r_m$, where $T^r_m$
are copies of $T_m$ in disjoint languages $\{K^r_m\}$,
$r\in\omega$. Indeed, disjoint predicates $K^r_m$ producing ${\rm
ar}(T_m)=m$ witness that ${\rm ar}(T_{mm})=m$. Finitely many these
predicates can not define all definable sets for $T_{mm}$ since
there are infinitely many of them. Thus, ${\rm aar}(T_{mm})=m$,
too.

Now for any $m<n$ we form $T_{mn}=T_n\sqcup
\bigsqcup\limits_{r\in\omega}T^r_m$. By $T_n$ in $T_{mn}$ and
$m<n$ we have ${\rm ar}(T_{mn})=n$. And ${\rm aar}(T_{mn})=m$
since there are infinitely many disjoint predicates of arity $m$.
\endproof

\medskip
The following theorem shows that all admissible pairs are
realized.

\begin{theorem}\label{th_admis}
For any admissible pair $(m,r)$ and $n\in\omega\setminus\{0,1\}$
there is a theory $T$ with ${\rm aar}(T)=n$ and ${\rm deg}_{\rm
ar}(T)=(m,r)$.
\end{theorem}

Proof. The admissible pair $(0,0)$ with ${\rm ar}(T)=n$ is
realized by Theorem \ref{th_aar_mn}. Now for an admissible pair
$(m,r)\ne(0,0)$ we can take a disjoint union $T$ of countably many
dense $n$-spherically ordered theories and of $m$ dense
$r$-spherically ordered theories. Using arguments for Theorem
\ref{th_aar_mn} we obtain ${\rm aar}(T)=n$ and ${\rm deg}_{\rm
ar}(T)=(m,r)$.~$\Box$

\begin{proposition}\label{praar1}
Any almost $n$-ary theory $T$ is $k$-ary for some $k\geq n$.
\end{proposition}

Proof. Let $T$ be an almost $n$-ary theory witnessed by the
formulae $\varphi_1(\overline{x}),\ldots,$
$\varphi_m(\overline{x})$. Then taking the set $\Delta_k$ of all
formulae with $k={\rm max}\{n,l(\overline{x})\}$ we observe, using
$\varphi_1(\overline{x}),\ldots,\varphi_m(\overline{x})$, that $T$
is $\Delta_k$-based, i.e., $T$ is $k$-ary. \endproof

\begin{corollary}\label{coraar1}
Any theory $T$ is $n$-ary for some $n$ iff $T$ is almost $m$-ary
for some $m$.
\end{corollary}

\begin{corollary}\label{coraar2}
Any theory $T$ is $n$-aritizable for some $n$ iff $T$ is almost
$m$-aritizable for some $m$.
\end{corollary}

\section{$\omega$-categorical almost $n$-ary and $n$-aritizable theories}

\begin{proposition}\label{praar_cat1}
If $T$ is an almost $n$-ary $\omega$-categorical theory, for some
$n$, then $T$ is almost $k$-ary for any
$k\in\omega\setminus\{0\}$, i.e., ${\rm aar}(T)=1$.
\end{proposition}

Proof. If $k\geq n$ then $T$ is almost $m$-ary, as noticed above.
If $k<n$ then we collect in a set $Z$ all formulae
$\varphi_1(\overline{x}),\ldots,\varphi_m(\overline{x})$
witnessing the almost $n$-arity of $T$ and, by Ryll-Nardzewski
Theorem, all non-equivalent formulae
$\psi_1(\overline{y}),\ldots,\psi_r(\overline{y})$ with
$l(\overline{y})=n$. Clearly, the set $Z$ witnesses that $T$ is
almost $k$-ary. Taking $k=1$ we obtain ${\rm aar}(T)=1$.~\endproof

\begin{corollary}\label{coraar3}
For any $\omega$-categorical theory $T$ either ${\rm aar}(T)=1$
with ${\rm ar}(T)\in\omega$, or ${\rm aar}(T)=\infty$ with ${\rm
ar}(T)=\infty$.
\end{corollary}

The following example illustrates Corollary \ref{coraar3}.

\begin{example}
{\rm Taking a dense linear order $K_2$ without endpoint we can
step-by-step extend it to a chain of dense $n$-spherical orders
$K_n$, $n\geq 2$, in the following way.

We put $(a,b,c)\in K_3$ if $(a,b)\in K_2$ and $(b,c)\in K_2$, or
$(b,c)\in K_2$ and $(c,a)\in K_2$, or $(c,a)\in K_2$ and $(a,b)\in
K_2$. If $K_n$, $n\geq 3$, is defined then we put
$(a_1,\ldots,a_{n+1})\in K_{n+1}$ if $(a_1,\ldots,a_n)\in K_n$ and
$(a_2,\ldots,a_{n+1})\in K_n$, or $a_2,\ldots,a_{n+1}\in K_n$ and
$(a_3,\ldots,a_{n+1},a_1)\in K_n$, or $(a_3,\ldots,a_{n+1},a_1)\in
K_n$ and $(a_4,\ldots,a_{n+1},a_1,a_2)\in K_n$. The obtained
structure $\mathcal{M}_\infty$ in the language
$\Sigma_\infty=\{K_n\mid n\in\omega\setminus\{0,1\}\}$ has an
$\omega$-categorical theory $T_\infty$ with ${\rm
aar}(T_\infty)={\rm ar}(T_\infty)=\infty$, since each new $K_n$
increases the arity. The same characteristics have restrictions of
$T_\infty$ to any infinite sublanguages.

At the same time each restriction $\mathcal{M}$ of
$\mathcal{M}_\infty$ to a finite nonempty sublanguage
$\{K_{n_1},\ldots,$ $K_{n_m}\}$ produces a theory $T$ with ${\rm
aar}(T)=1$ with $${\rm ar}(T)={\rm max}\{n_1,\ldots,n_m\}.$$}
\end{example}

By Corollary \ref{coraar3} and the definition of almost
aritizability we immediately have:

\begin{corollary}\label{coraar4}
Any restriction $T$ of $n$-ary $\omega$-categorical theory $T'$ is
almost unary-tizable.
\end{corollary}

\section{Operations for almost $n$-ary and $n$-aritizable theories}

In this section we consider links for arities of theories with
respect to disjoint unions of theories and $E$-definable
compositions of theories.

\begin{theorem}\label{thduaa}
$1.$ For any theories $T_1$, $T_2$ and their disjoint union
$T_1\sqcup T_2$, both $T_1$ and $T_2$ are almost $n$-ary iff
$T_1\sqcup T_2$ is almost $n$-ary, moreover, ${\rm aar}(T_1\sqcup
T_2)={\rm max}\{{\rm aar}(T_1),{\rm aar}(T_2)\}$.

$2.$ For any theories $T_1$, $T_2$ and their disjoint union
$T_1\sqcup T_2$, both $T_1$ and $T_2$ are almost $n$-aritizable
iff $T_1\sqcup T_2$ is almost is $n$-aritizable.
\end{theorem}

Proof. 1. Let $\Phi_1$ and $\Phi_2$ be finite sets of formulas
witnessing that $T_1$ and $T_2$ are almost $n$-ary, respectively.
Using the definition of disjoint union and Theorem \ref{thdu} we
obtain that the finite set $\Phi_1\cup \Phi_2$ witnesses that
$T_1\sqcup T_2$ is almost $n$-ary. Conversely, if a finite set
$\Phi$ of formulas witnesses that $T_1\sqcup T_2$ is almost
$n$-ary then $Phi$ witnesses that both $T_1$ and $T_2$ are almost
$n$-ary.

The equality ${\rm aar}(T_1\sqcup T_2)={\rm max}\{{\rm
aar}(T_1),{\rm aar}(T_2)\}$ follows from the definition of
disjoint union since if ${\rm aar}(T_1\sqcup T_2)=k$ then the
maximal value of ${\rm aar}(T_1)$ and ${\rm aar}(T_2)$ is
responsible for this equality.

Item 2 follows from Item 1 since expansions of $T_1$ and $T_2$
correspond expansions of $T_1\sqcup T_2$: some expansions of
$T'_1$ and $T'_2$ of $T_1$ and $T_2$, respectively, are almost
$n$-ary iff $T'_1\sqcup T'_2$ produces an almost $n$-ary expansion
of $T_1\sqcup T_2$. \endproof

\medskip
Using induction we obtain the following:

\begin{corollary}\label{coduaa}
$1.$ For any theories $T_1, T_2,\ldots,T_m$ and their disjoint
union $\bigsqcup\limits_{i=1}^m T_i$, all $T_1, T_2,$ $\ldots,T_m$
are almost $n$-ary iff $\bigsqcup\limits_{i=1}^m T_i$ is almost
$n$-ary, moreover, $${\rm aar}\left(\bigsqcup\limits_{i=1}^m
T_i\right)={\rm max}\{{\rm aar}(T_i)\mid i\leq m\}.$$

$2.$ For any theories $T_1, T_2,$ $\ldots,T_m$ and their disjoint
union $\bigsqcup\limits_{i=1}^m T_i$, all $T_1, T_2,\ldots,T_m$
are almost $n$-aritizable iff $\bigsqcup\limits_{i=1}^m T_i$ is
almost $n$-aritizable.
\end{corollary}

\begin{note}{\rm Both almost $n$-arity and almost
$n$-aritizability can fail taking disjoint unions of infinitely
many theories $T_i$, $i\in I$. Indeed, each theory $T_i$ can have
its own finite set $\Phi_i$ of formulas witnessing the almost
$n$-arity/$n$-aritizability, say in disjoint languages, whereas
finite unions $\bigcup\Phi_i$ can not witness the almost
$n$-arity/$n$-aritizability for $\bigsqcup\limits_{i\in I}T_i$.}
\end{note}

Generalizing Theorem \ref{thcomp} we obtain:

\begin{theorem}\label{thcompaa} $1.$ For any theories $T_1$ and $T_2$ and
their $E$-definable composition $T_1[T_2]$, $T_1$ and $T_2$ are
almost $n$-ary iff $T_1[T_2]$ is almost $n$-ary, moreover, ${\rm
aar}(T_1[T_2])={\rm max}\{{\rm aar}(T_1),{\rm aar}(T_2),2\}$, if
models of $T_1$ and of $T_2$ have at least two elements, and ${\rm
aar}(T_1[T_2])={\rm max}\{{\rm aar}(T_1),{\rm aar}(T_2)\}$, if a
model of $T_1$ or $T_2$ is a singleton.

$2.$ For any theories $T_1$ and $T_2$ and their $E$-definable
composition $T_1[T_2]$, $T_1$ and $T_2$ are almost $n$-aritizable
iff $T_1[T_2]$ is almost $n$-aritizable.
\end{theorem}

Proof. 1. Let $T_i$ be $\Delta_i$-based for $i=1,2$. Since
$T_1[T_2]$ is $E$-definable it is $\Delta$-based, where $\Delta$
consists of formulae in $\Delta_1\cup\Delta_2$ and $E(x,y)$
\cite{EKS20}. Now assuming that $T_1$ and $T_2$ are almost $n$-ary
we can choose $\Delta_i$ consisting of $n$-formulae and finitely
many formulae forming $\Phi_i$, $i=1,2$. Hence $T_1[T_2]$ is
almost $n$-ary.

Conversely, if $T_1[T_2]$ is almost $n$-ary and it is witnessed by
a set $\Phi$ of formulae then $\Phi$ witnesses that $T_1$ and
$T_2$ are almost $n$-ary.

If models of $T_1$ and of $T_2$ have at least two elements then
$T_1[T_2]$ is at least binary that witnessed by the formula
$E(x,y)$. Thus since $T_1[T_2]$ is $\Delta$-based we have ${\rm
aar}(T_1[T_2])={\rm max}\{{\rm aar}(T_1),{\rm aar}(T_2),2\}$. If
$T_1$ or $T_2$ is a theory of singleton then $T_1[T_2]$ is
$(\Delta_1\cup\Delta_2)$-based implying ${\rm aar}(T_1[T_2])={\rm
max}\{{\rm aar}(T_1),{\rm aar}(T_2)\}$.

Item 2 follows from Item 1 repeating the arguments for Item 2 of
Theorem \ref{thduaa}. \endproof

\medskip
Theorem \ref{thcompaa} immediately implies

\begin{corollary}\label{cocompaa}
Any composition of finitely many almost $n$-ary {\rm (}almost
$n$-aritizable{\rm )} theories, for $n\geq 2$, is again many
almost $n$-ary {\rm (}almost $n$-aritizable{\rm )}.
\end{corollary}

\section{Conclusion}

We considered possibilities for almost arities and almost
aritizabilities of theories and their dynamics both in general
case, for $\omega$-categorical theories, and with respect to
operations for theories. It would be interesting to describe
values of almost arities and almost aritizabilities for natural
classes of theories.


\begin{thebibliography}{99}
\bibitem{aafot} S.V.~Sudoplatov, {\it Arities and aritizabilities of first-order
theories}, Siberian Electronic Mathematical Reports (to appear),
arXiv:2112.09593v1 [math.LO], 2021. 15~pp.

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

\bibitem{CCMCT} S.V.~Sudoplatov, {\it Classification of Countable Models of
Complete Theories}, No\-vo\-si\-birsk~: NSTU, 2018.

\bibitem{KulMac} B.Sh.~Kulpeshov, H.D.~Macpherson, {\it Minimality
conditions on circularly ordered structures}, Mathematical Logic
Quarterly, {\bf 51}:4 (2005), 377--399.

\bibitem{Kulp3} A.B.~Altaeva, B.Sh.~Kulpeshov, {\it On almost binary
weakly circularly minimal structures}, Bulletin of Karaganda
University, Mathematics, {\bf 78}:2 (2015), 74--82.

\bibitem{Kulp4} B.Sh.~Kulpeshov, {\it On almost binarity in weakly
circularly minimal structures}, Eurasian Mathematical Journal,
{\bf 7}:2 (2016), 38--49.

\bibitem{EKS20} D.Yu.~Emelyanov, B.Sh.~Kulpeshov, S.V.~Sudoplatov,
{\it Algebras of binary formulas for compositions of theories},
Algebra and Logic, {\bf 59}:4 (2020), 295--312.

\bibitem{Wo} R.E.~Woodrow, {\it Theories with a finite number of
countable  models and a small language}, Ph.~D. Thesis, Simon
Fraser University, 1976.

\end{thebibliography}


\end{document}
