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\title[Properties of ranks]{Properties of ranks for families of strongly minimal theories}
\author{B.Sh. Kulpeshov, S.V. Sudoplatov}%
\address{Beibut Shaiykovich Kulpeshov
\newline\hphantom{iii} Kazakh British Technical University
\newline\hphantom{iii} Tole bi street, 59
\newline\hphantom{iii} 050000, Almaty, Kazakhstan.
}%
\email{b.kulpeshov@kbtu.kz}%
\address{Novosibirsk State Technical University
\newline\hphantom{iii} K. Marx avenue, 20
\newline\hphantom{iii} 630073, Novosibirsk, Russia.}
\email{kulpeshov@corp.nstu.ru}%

% .
%, , , ; .

\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{sudoplatov@corp.nstu.ru}%
\address{Sobolev Institute of Mathematics,
\newline\hphantom{iii} Academician Koptyug avenue, 4
\newline\hphantom{iii} 630090, Novosibirsk, Russia.
}%
\email{sudoplat@math.nsc.ru}

\thanks{\sc Kulpeshov, B.Sh., Sudoplatov, S.V.,
Properties of ranks for families of strongly minimal theories}
\thanks{\copyright \ 2022 Kulpeshov B.Sh., Sudoplatov S.V.}
\thanks{\rm The research is supported by Russian Scientific
Foundation, Project No.~22-21-00044.}
\thanks{\it Received  December, ??, 2021, published  ???, ??,  2022.}%


\semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} We study rank properties for families of
strongly minimal theories. A criterion for $e$-total
transcendence of families of strongly minimal theories is
obtained in terms of the description of the language
symbols.\medskip

\noindent{\bf Keywords:} strongly minimal theory, sentence, family of
theories, rank for a fa\-mi\-ly of theories.
 \end{quote}
}


\section{Introduction}

In the study of elementary theories, an important role is played
by the description of their interrelationships and derived
objects, including main characteristics. Essen\-t\-i\-al
interrelationships between theories arise when considering their
various com\-bi\-na\-t\-ions \cite{cs, ccct}, closures and
generating sets for families of theories \cite{cl, lut},
appro\-x\-i\-ma\-t\-ions of theories \cite{at}. Along with
combinations of theories in general, various combinations of
ordered theories are considered, which allow describing their
esse\-n\-t\-i\-al structural properties \cite{AKS, KS_P1, KS_P2,
KS_E}. Among various approximations of theories, one of the
central places is occupied by approximations by theories of finite
structures, which define pseudofinite structures \cite{Rosen, Vaa,
ChHr, MS}. Important derived objects are algebras of definable
subfamilies \cite{dst, MS19}. Main characteristics related to the
families of theories are the $e$-spectra \cite{cs, rest}, the
ranks for the families of theories \cite{dst, RSrank, MarkhSud}
and for the formulas \cite{FPM, PS21}. Rank values allow one to
determine how complex certain types of families of theories are,
both in general case and in relation to natural classes of
theories, including classes of substitution theories \cite{M19},
theories of abelian groups \cite{PS19}, ordered theories
\cite{KS_rank2019, KPS_2021}.

The present article is concerned the notion of {\it strong
minimality} initially deeply investigated in \cite{BaLa}. We study
rank properties for families of strongly minimal theories. A
criterion for $e$-total transcendence of families of strongly
minimal theo\-ries is obtained in terms of the description of the
language symbols.

\section{Preliminary notions}

\begin{definition} {\rm \cite{BaLa} Let $T$ be a complete theory without finite models.
The theory $T$ is said to be {\it strongly minimal} if for any model
$\mathcal{M}\models T$, any formula $\phi(x,\bar y)$ and any tuple
$\bar a\in M$ either the set $\phi(\mathcal{M},\bar a)$ is finite
or the set $\neg\phi(\mathcal{M},\bar a)$ is finite.}
\end{definition}

\begin{definition}\rm \cite{RSrank} Let $\mathcal{T}$ be a family of complete theories in a fixed
language $\Sigma$, $\phi$ be an arbitrary $\Sigma$-sentence. Then
the set $\mathcal{T}_{\phi}:=\{T\in\mathcal{T}\mid T\models
\phi\}$ is said to be the {\it $\phi$-neighbourhood} of the family
$\mathcal{T}$, or the {\it $s$-definable} subfamily of
$\mathcal{T}$ defined by the sentence~$\phi$.

We define the {\em rank} ${\rm RS}$ for the family $\mathcal{T}$
as follows:

(1) ${\rm RS}(\mathcal{T})=-1$, if $\mathcal{T}=\emptyset$.

(2) ${\rm RS}(\mathcal{T})=0$, if  $\mathcal{T}$ is a finite
nonempty family.

(3) ${\rm RS}(\mathcal{T})\ge 1$, if $\mathcal{T}$ is infinite.

(4) ${\rm RS}(\mathcal{T})\ge \alpha+1$, if there are pairwise
inconsistent $\Sigma$-sentences $\phi_n, n\in\omega$, such that
${\rm RS}(\mathcal{T}_{\phi_n})\ge \alpha$.

(5) If $\delta$ is a limit ordinal, then ${\rm RS}(\mathcal{T})\ge
\delta$, if  ${\rm RS}(\mathcal{T})\ge \beta$ for any
$\beta<\delta$.

We put ${\rm RS}(\mathcal{T})=\alpha$, if ${\rm
RS}(\mathcal{T})\ge\alpha$ and $\neg[{\rm
RS}(\mathcal{T})\ge\alpha+1]$.

If ${\rm RS}(\mathcal{T})\ge\alpha$ for any $\alpha$, then we put
${\rm RS}(\mathcal{T})=\infty$.

A family $\mathcal{T}$ is called {\em $e$-totally transcendental},
or {\em totally transcendental}, if ${\rm RS}(\mathcal{T})$ is an
ordinal.

If a family $\mathcal{T}$ is $e$-totally transcendental, with
${\rm RS}(\mathcal{T})=\alpha\geq 0$, then the {\em degree} ${\rm
ds}(\mathcal{T})$ of $\mathcal{T}$ is the maximal number of
pairwise inconsistent sentences $\phi_i$ with ${\rm
RS}(\mathcal{T}_{\varphi_i})=\alpha$.
\end{definition}

By the definition, if ${\rm RS}(\mathcal{T})=\alpha\in{\rm Ord}$
then ${\rm ds}(\mathcal{T})\in\omega\setminus\{0\}$.

\section{Families of strongly minimal theories and their ranks}

Throughout this article we denote by $\mathcal{T}_{\Sigma}$ the family of all
strongly minimal theories in a language $\Sigma$.

\begin{proposition}\label{prop1} Let $\Sigma^1_{\kappa}:=\{P^1_i\}_{i<\kappa}$, where
$\kappa$ is a cardinal. Then the following holds:

{\rm (1)} If $\kappa<\omega$ then ${\rm
RS}(\mathcal{T}_{\Sigma^1_{\kappa}})=\kappa$.

{\rm (2)} If $\kappa \ge \omega$ then ${\rm
RS}(\mathcal{T}_{\Sigma^1_{\kappa}})=\infty$.
\end{proposition}

\begin{proof}
(1) By virtue of strong minimality, each unary predicate selects a
finite or cofinite set in any model of an arbitrary strongly
minimal theory.

We construct an infinitely branching tree of length $\kappa$ for the
family $\mathcal{T}_{\Sigma^1_{\kappa}}$ of theories. At the first
level, we distinguish theories in which the first predicate $P_1$
selects in any model a finite set consisting of $n$ elements for
each $n <\omega$. Further, at the $i$-th level ($i\le\kappa$), we
distinguish theories in which the predicate $P_i$ selects a finite
set consisting of $n$ elements for each $n<\omega$. Therefore, we
obtain ${\rm RS}(\mathcal{T}_{\Sigma^1_{\kappa}})=\kappa$.

(2) Since there exist infinitely many unary predicates, then one
can construct an infinitely branching tree of infinite length,
whence ${\rm RS}(\mathcal{T}_{\Sigma^1_{\kappa}})=\infty$.
\end{proof}

Let $\Sigma^f_1:=\{f^1\}$, where $f$ is a unary functional symbol,
i.e. any theory $T$ of the language $\Sigma^f_1$ satisfies
the following sentence: $T\models (\forall x)(\exists! y) f(x)=y.$

Let $T$ be a complete theory of the language $\Sigma^f_1$,
$M\models T$.

We say that an element $a\in M$ {\it forms a cycle of length $1$}
if $f(a)=a$.

We say that elements $a_1, a_2, \ldots, a_n\in M$ {\it form a
cycle of length $n$} if $a_i\ne a_j$ for any $1\le i,j \le n$ with
$i\ne j$, $f(a_1)=a_2, f(a_2)=a_3, \ldots, f(a_{n-1})=a_n$ and
$f(a_n)=a_1$.

Consider the following sentences:
$$\theta^1_n:=\exists x_1 \ldots \exists x_n [\land_{i\ne j} x_i\ne x_j \land \land_{i=1}^{n-1} f(x_i)=x_{i+1} \land f(x_n)=x_1\land$$
$$\forall y_1 \ldots \forall y_n (\land_{i\ne j} y_i\ne y_j \land \land_{i=1}^{n-1} f(y_i)=y_{i+1}\land f(y_n)=y_1 \to \lor_{i=1}^n y_1=x_i)],$$
$$\theta^2_n:=\exists x_1 \ldots \exists x_n \exists y_1 \ldots \exists y_n[\land_{i\ne j} x_i\ne x_j \land \land_{i=1}^{n-1} f(x_i)=x_{i+1} \land f(x_n)=x_1\land$$
$$\land_{i\ne j} y_i\ne y_j \land \land_{i=1}^{n-1} f(y_i)=y_{i+1}\land f(y_n)=y_1\land$$
$$ \forall z_1 \ldots \forall z_n (\land_{i\ne j} z_i\ne z_j \land \land_{i=1}^{n-1} f(z_i)=z_{i+1}\land f(z_n)=z_1 \to \lor_{i=1}^n (z_1=x_i\lor z_1=y_i))].$$

The sentence $\theta^1_n$ asserts that there exists a unique cycle of
length $n$, and the sentence $\theta^2_n$ asserts that there exist
exactly two cycles of length $n$. Similarly, we can construct a
sentence $\theta^k_n$ asserting that there exist exactly $k$ cycles
of length $n$, for any $k,n<\omega$.

\begin{proposition}\label{prop2} Let $\Sigma^f_1:=\{f^1\}$, where $f$ is a unary functional symbol.
Then ${\rm RS}(\mathcal{T}_{\Sigma^f_1})=\infty$.
\end{proposition}

\begin{proof}
We construct an infinitely branching tree of infinite length for
$\mathcal{T}_{\Sigma^f_1}$.

At the first level, we distinguish theories in which there are
exactly $n$ cycles of length $1$ for each $n<\omega$. At the
second level, we distinguish theories in which there are exactly
$n$ cycles of length $2$ for each $n<\omega $. At the $k$-th level
($k<\omega$), we distinguish theories in which there are exactly
$n$ cycles of length $k$ for each $n <\omega $. Thus, we obtain
${\rm RS}(\mathcal{T}_{\Sigma^f_1})=\infty$.
\end{proof}

Since $k$-ary predicates and $k$-ary operations, for $k\geq 2$,
allow to interpret unary operations, Proposition \ref{prop2}
implies:

\begin{corollary}\label{cor01} If a language $\Sigma$ contains a $k$-ary predicate symbol or a $k$-ary fun\-c\-tio\-nal symbol, where $k\geq 2$,
then ${\rm RS}(\mathcal{T}_{\Sigma})=\infty$.
\end{corollary}

\begin{proposition}\label{propMS} {\rm \cite{MarkhSud}} If $\Sigma$ is a language consisting of constant symbols then for the family $\mathcal{T}$ of all theories
in the language $\Sigma$ either ${\rm RS}(\mathcal{T})=1$ and
${\rm ds}(\mathcal{T})=P(n)$, where $P(n)$ is the number of
partitions of $n$-element sets, if $\Sigma$ consists of
$n$ symbols, where $n<\omega$, or ${\rm RS}(\mathcal{T})=\infty$, if
$\Sigma$ has infinitely many symbols.
\end{proposition}

Proposition \ref{propMS} implies:

\begin{fact}\label{fact1} Let $\Sigma^c_{\kappa}:=\{c_i\}_{i<\kappa}$, where $\kappa$ is a cardinal.
Then ${\rm RS}(\mathcal{T}_{\Sigma^c_{\kappa}})=0$ and ${\rm
ds}(\mathcal{T}_{\Sigma^c_{\kappa}})=P(\kappa)$, where $P(\kappa)$
is the number of partitions of $n$-element set, if
$\kappa<\omega$, or ${\rm
RS}(\mathcal{T}_{\Sigma^c_{\kappa}})=\infty$, if
$\kappa\geq\omega$.
\end{fact}


\begin{proposition}\label{prop3}
Let $\Sigma^{1, c}_{k,m}:=\{P^1_1, \ldots, P^1_k, c_1, \ldots,
c_m\}$, where $k, m<\omega$. Then ${\rm
RS}(\mathcal{T}_{\Sigma^{1, c}_{k,m}})=k$ and ${\rm
ds}(\mathcal{T}_{\Sigma^{1, c}_{k,m}})=P'(k,m)$, where $P'(k,m)$
is the total number of actions by predicates $P^1_1, \ldots,
P^1_k$ on each of $P(m)$ partitions of the set
$\{c_1,\ldots,c_m\}$ on $r$ parts, $1\leq r\leq m$:
$P'(k,m)=\sum\limits_{r}2^{kr}.$
\end{proposition}

\begin{proof}
By virtue of Proposition \ref{prop1} and Fact \ref{fact1}, we
assert that ${\rm RS}(\mathcal{T}_{\Sigma^{1, c}_{k,m}})=k$. Let
us now understand that ${\rm ds}(\mathcal{T}_{\Sigma^{1,
c}_{k,m}})=2^{km}$. Each of the unary predicates $P^1_i$ can act
on the set of identified constants $c_j$ in two ways, i.e. either
$P^1_i(c_j)$, for all identified constants $c_j$, or $\neg
P^1_i(c_j)$ (each constant has only two possibilities for a chosen
unary predicate). Therefore, taking an arbitrary partition of the
set $\{c_1,\ldots,c_m\} $ into $r$ parts, we obtain the
multiplication $r$ times $2^k$ of the actions of the predicates
$P^1_1,\ldots,P^1_k$ on each element partition equal to $2^{kr}$.
Running through all the partitions of constants and all variants
of the influence of the predicates $P^1_1,\ldots,P^1_k $ on the
elements of the partitions of the set $\{c_1,\ldots,c_m\} $, we
obtain the desired equality ${\rm ds}(\mathcal{T}_{\Sigma^{1,
c}_{k,m}})=P'(k,m)$.
\end{proof}

\begin{proposition}\label{propMS2} {\rm \cite{MarkhSud}} If $\Sigma$
is a language of $0$-ary predicates, then for the family
$\mathcal{T}$ of all theories in the language $\Sigma$ either
${\rm RS}(\mathcal{T})=0$ and ${\rm ds}(\mathcal{T})=2^m$, if
$\Sigma$ consists of $m$ $0$-ary predicates, where $m<\omega$, or ${\rm
RS}(\mathcal{T})=\infty$, if $\Sigma$ has infinitely many symbols.
\end{proposition}

Proposition \ref{propMS2} implies:

\begin{fact}\label{fact2} Let $\Sigma^0_{\kappa}:=\{P^0_i\}_{i<\kappa}$,
where $\kappa$ is a cardinal. Then either ${\rm
RS}(\mathcal{T}_{\Sigma^0_{\kappa}})=0$ and ${\rm
ds}(\mathcal{T}_{\Sigma^c_{\kappa}})=2^\kappa$, if
$\kappa<\omega$, or ${\rm
RS}(\mathcal{T}_{\Sigma^0_{\kappa}})=\infty$, if
$\kappa\geq\omega$.
\end{fact}

In view of Fact \ref{fact2} if we add to the language $\Sigma$
finitely many $0$-ary predicate symbols then we preserve the
$e$-total transcendence of the family of all strongly minimal
theories in the expanded language $\Sigma'$, whereas adding to the
language $\Sigma$ infinitely many $0$-ary predicate symbols we
obtain the family $\mathcal{T}_\Sigma'$ with ${\rm
RS}(\mathcal{T}_\Sigma')=\infty$. Thus, Propositions \ref{prop1},
\ref{prop2}, \ref{prop3}, Corollary \ref{cor01}, and Facts
\ref{fact1}, \ref{fact2} imply the following:

\begin{theorem}\label{Thm} A family $\mathcal{T}_{\Sigma}$ of strongly minimal theories is $e$-totally transcendental if and only if the language $\Sigma$
consists of finitely many constant symbols as well as of finitely
many $0$-ary and unary predicate symbols.
\end{theorem}

\begin{note}
{\rm Assuming that distinct constant symbols are interpreted by
distinct elements, $e$-totally transcendental families of strongly
minimal theories in Theorem \ref{Thm} admit arbitrarily many
constant symbols in the language.}
\end{note}

\begin{note}
{\rm Since ${\rm RS}(\mathcal{T})\leq{\rm RS}(\mathcal{T}')$ for
$\mathcal{T}\subseteq\mathcal{T}'$ then the criterion for
$e$-total transcendence of families of strongly minimal theories
in Theorem \ref{Thm} can be trans\-formed for any family
$\mathcal{T}'$ containing the family $\mathcal{T}_{\Sigma}$ of
strongly minimal theories of given language $\Sigma$. In
particular, it holds for families of $\omega$-stable, superstable,
and stable theories.}
\end{note}


\bigskip
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\end{document}
