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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
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\begin{document}
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\title[Spherical orders]{Spherical orders, properties and countable spectra of their 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 Committee of Science in Science and Higher Education 
Ministry of the Republic of Kazakhstan, Grant No.
AP19674850. The work of the second 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  November, ??, 2022, published  ???, ??,  2023.}%


\semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} We study semantic and syntactic
properties of spherical orders and their elementary theories,
including finite and dense orders and their theories. It is shown
that theories of dense $n$-spherical orders are countably
categorical and decidable. The values for spectra of countable
models of unary expansions of $n$-spherical theories are
described. The Vaught conjecture is confirmed for countable
constant expansions of dense $n$-spherical theories.\medskip

\noindent{\bf Keywords:} spherical order, elementary theory, dense
spherical order, countably categorical theory, spectrum of
countable models, Vaught con\-jec\-tu\-re.
 \end{quote}
}



\section{Introduction}\label{sec1}

Spherical orders \cite{aafot, ananat} are natural generalizations
of linear and circular orders \cite{KulMac, Kulp3, Kulp4} allowing
to compare tuples and their elements on $n$-spheres. These orders
present geometric realizations of arbitrary arities of formulae
along with known algebraic ones \cite{Semenov2003}.

In the paper, we study some kinds of $n$-spherical orders
including finite and dense ones, and their elementary theories. We
show that theories of dense $n$-spherical orders are countably
categorical and decidable. Based on classical Ehrenfeucht
examples, we construct Ehrenfeucht expansions of dense
$n$-spherical orders pro\-du\-c\-ing theories with arbitrarily
finitely many $m\ne 0,2$ countable models. The Vaught conjecture
is confirmed for countable constant expansions of the dense
$n$-spherical theories.

We deals with complete first-order theories and use without
specifications no\-tions, notations and results on arities of
theories \cite{aafot, ananat}, on circular orders \cite{KulMac,
Kulp3, Kulp4} and on distributions of countable models of complete
theories \cite{CCMCT}.

\section{Preliminaries}\label{sec2}

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

\medskip
The following lemma, being a corollary of Compactness Theorem, was
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}

Lemma \ref{lem01} allows to reduce, in general, possibilities of
$\Delta$-basedness of a theory $T$ to the $\Delta$-deducibility of
its types, and vice versa.

Below we consider special forms of $\Delta$-basedness.

\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 $T$-formulae, each of which is of one free variable, and of
formulae 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 $T$-formulae, 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 admit the case $n=0$ for $n$-formulae $\varphi(\overline{x})$,
too. In such a case $\varphi(\overline{x})$ is just $T$-equivalent
to a sentence $\forall\overline{x}\varphi(\overline{x})$.

We say that the {\em arity} of $T$ is $n$ and write ${\rm
ar}(T)=n$ for this if $T$ is $n$-ary but not $(n-1)$-ary. 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 ${\rm ar}_T(\varphi)$.

By the definition, ${\rm ar}(\varphi)\leq\vert{\rm
FV}(\varphi)\vert$, where ${\rm FV}(\varphi)$ is the set of all
free variables of the formula~$\varphi$.


\section{$n$-spherical orders}\label{sec3}

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) $\forall x\forall y \forall z (K_3(x,y,z)\land K_3(y,x,z)
 \leftrightarrow x\approx y \lor y\approx z \lor z\approx x);$\\
 (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) $\forall x\forall y \forall z (K_3(x,y,z)\lor K_3(y,x,z)).$

By the definition, ${\rm ar}(K_3(x,y,z))=3$ if the relation has at
least three element domain, where the {\em domain} is the set of
projections of $K_3$ into its some/any co\-or\-di\-na\-te. Hence,
theories whose models have infinite circular order relations are
at least $3$-ary.

The following generalization of circular order produces an {\em
$n$-ball}, or {\em $n$-spherical}, or {\em $n$-circular} order
relation \cite{aafot, ananat}, for $n\geq 4$, which is described
by an $n$-ary relation
$K_n$ satisfying the following conditions:\\
 (nso1) $\forall x_1,\ldots,x_n (K_n(x_1,x_2,\ldots,x_n)\to K_n(x_2,\ldots,x_n,x_1));$\\
 (nso2) $\forall x_1,\ldots,x_n
 \biggm((K_n(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)\land$
 $$\land K_n(x_1,\ldots,x_j,\ldots,x_i,\ldots,x_n))\leftrightarrow\bigvee\limits_{1\leq k<l\leq n} x_k\approx x_l\biggm)$$ for any $1\leq i<j\leq n$;\\
 (nso3) $\forall x_1,\ldots,x_n\Biggm(K_n(x_1,\ldots,x_n)\to$
 $$\to\forall t\left(\bigvee\limits_{i=1}^nK_n(x_1,\ldots,x_{i-1},t,x_{i+1},\ldots,x_n)\right)\Biggm);$$\\
 (nso4) $\forall x_1,\ldots,x_n (K_n(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)\lor$
 $$\lor K_n(x_1,\ldots,x_j,\ldots,x_i,\ldots,x_n)),\,\, 1\leq i<j\leq
 n.$$

Structures $\mathcal{A}=\langle A,K_n\rangle$ with $n$-spherical
orders $K_n$ will be called {\em $n$-spherically ordered sets}, or
{\em $n$-spherical orders}, too.

\begin{note} {\rm
By the axiom (nso2) the universe $A$ of a structure
$\mathcal{A}=\langle A,K_n\rangle$ with an $n$-spherical order
$K_n$ is uniquely defined as the set of all coordinates of tuples
in~$K_n$. Moreover, by (nso1) it is covered by a set of
coordinates of tuples in $K_n$ with a fixed number, say $A$ is
represented as the set of second coordinates of tuples in~$K_n$.

Similarly to the circular order, the universe $A$ is the {\em
domain} for the $n$-spherical order $K_n$.}
\end{note}

%\medskip
\begin{note} {\rm The axioms above are naturally adapted for $n=2$ producing a linear
order $K_2$. Here (nso2) produces the reflexivity: $\forall x
K_2(x,x)$, and the antisymmetry: $\forall
x_1,x_2(K_2(x_1,x_2)\wedge K_2(x_2,x_1)\to x_1\approx x_2)$,
(nso1) is replaced by the transitivity: $$\forall
x_1,x_2,x_3(K_2(x_1,x_2)\wedge K_2(x_2,x_3)\to K_2(x_1,x_3)),$$
and the axioms (nso3) and (nso4) produce the linearity: $$\forall
x_1,x_2(K_2(x_1,x_2)\vee K_2(x_2,x_1)).$$

The only case $n=2$, i.e. a linear order, can admit endpoints,
since for the cases $n\geq 3$ each element lays between other
ones.}
\end{note}

Now we consider a series of models illustrating $n$-spherical
orders.

\begin{example}\label{ex12} {\rm Recall that a {\em directed tetrahedron}, or {\em
dirtetrahedron} \cite{MP} is a $4$-tuple $(A_1,A_2,A_3,$ $A_4)$ of
points in $\mathbb R^3$ which do not belong to a common plane. For
the dirtetrahedron $(A_1,A_2,A_3,A_4)$ we define the structure
$\langle\{A_1,A_2,A_3,A_4\},K^4_4\rangle$, where the $4$-ary
relation $K^4_4$ on the $4$-element set $\{A_1,A_2,A_3,A_4\}$ is
the closure of the set of even permutations of $(A_1,A_2,A_3,A_4)$
with fixed $A_1$ by the cyclic permutation $(1,2,3,4)$ (as it
follows from Axiom nso1), united with the set $I_4$ of $4$-tuples
$(A_i,A_j,A_k,A_l)$ having at least two equal points (as it
follows from Axiom nso2). Thus $K^4_4$ is generated by three
dirtetrahedrons
$$(A_1,A_2,A_3,A_4), (A_1,A_3,A_4,A_2), (A_1,A_4,A_2,A_3)$$ using
cyclic permutations and $I_4$.

We immediately check that $K^4_4$ satisfies the axioms
(nso1)--(nso4) for $n=4$, producing a $4$-spherical order. It
consists of all $4$-tuples on $\{A_1,A_2,A_3,A_4\}$
de\-scri\-b\-ed above forbidding $\frac{4!}{2}$ possibilities for
odd permutations and producing $\vert K^4_4 \vert
=4^4-\frac{4!}{2}=4(4^3-3)=244$.

\begin{figure}
\begin{center}
\setlength{\unitlength}{1.1mm}
\begin{picture}(45,45)(4.5,8)

\qbezier(45.0, 30.0)(45.0, 38.2843) (39.1421, 44.1421)
\qbezier(39.1421, 44.1421)(33.2843, 50.0) (25.0, 50.0)
\qbezier(25.0, 50.0)(16.7157, 50.0) (10.8579, 44.1421)
\qbezier(10.8579, 44.1421)(5.0, 38.2843) (5.0, 30.0) \qbezier(5.0,
30.0)(5.0, 21.7157) (10.8579, 15.8579) \qbezier(10.8579,
15.8579)(16.7157, 10.0) (25.0, 10.0) \qbezier(25.0, 10.0)(33.2843,
10.0) (39.1421, 15.8579) \qbezier(39.1421, 15.8579)(45.0, 21.7157)
(45.0, 30.0)

\put(45,30){\makebox(0,0)[cc]{$\bullet$}}
\put(5,30){\makebox(0,0)[cc]{$\bullet$}}
\put(25,50){\makebox(0,0)[cc]{$\bullet$}}
\put(25,10){\makebox(0,0)[cc]{$\bullet$}}

\put(48,30){\makebox(0,0)[cc]{$a_1$}}
\put(25,52.5){\makebox(0,0)[cc]{$a_2$}}
\put(2.5,30){\makebox(0,0)[cc]{$a_3$}}
\put(25.7,7.5){\makebox(0,0)[cc]{$a_4$}}

\put(25.6,49.99){\vector(-4,0){0.5}}
\put(5,31){\vector(0,-4){0.5}} \put(24.2,10.01){\vector(4,0){0.5}}
\put(45,29){\vector(0,4){0.5}}

\put(45,30){\vector(-4,0){39.5}} \put(5,30){\vector(1,-1){19.5}}
\put(25,10){\vector(0,1){39.5}} \put(25,50){\vector(1,-1){19.5}}

\qbezier(45,30)(35,28)(25,10) \qbezier(25,10)(17,30)(25,50)
\qbezier(25,50)(13,34)(5,30) \qbezier(5,30)(25,38)(45,30)

\put(25.5,11){\vector(-1,-1){0.5}}
\put(24.05,46.85){\vector(1,4){0.5}}
\put(6.85,31.2){\vector(-3,-2){0.5}}
\put(43.3,30.6){\vector(3,-1){0.5}}

\vspace{-10mm}
\end{picture}
\end{center}
%\hspace{54mm}\parbox[t]{0.3\textwidth}\caption{} \label{fig111}
\caption{} \label{fig111}
\end{figure}

Replacing points $A_1,A_2,A_3,A_4$ by elements $a_1,a_2,a_3,a_4$,
respectively, we obtain an isomorphic copy for $K^4_4$ with
generating $4$-tuples represented in Figure \ref{fig111}. }
\end{example}

\begin{example}\label{ex13} {\rm
Similarly Example \ref{ex12}, taking an $n$-tuple
$(A_1,A_2,\ldots, A_n)$ of points in $\mathbb R^{n-1}$, $n\geq 5$,
which do not belong to a common hyperplane we obtain a {\em
directed $n$-hedron}. For the directed $n$-hedron
$(A_1,A_2,\ldots, A_n)$ we define the structure
$\langle\{A_1,A_2,$ $\ldots,A_n\},K^n_n\rangle$, where the $n$-ary
relation $K^n_n$ on the $n$-element set $\{A_1,A_2,\ldots,A_n\}$
is the closure of the set of even permutations of
$(A_1,A_2,\ldots, A_n)$ with fixed $A_1$ by the cyclic permutation
$(1,2,\ldots,n)$ (as it follows from Axiom nso1), united with the
set $I_n$ of $n$-tuples $(A_{i_1},A_{i_2},$ $\ldots, A_{i_n})$
having at least two equal points (as it follows from Axiom nso2).

We check that $K^n_n$ satisfies the axioms (nso1)--(nso4) in the
general case, pro\-du\-c\-ing an $n$-spherical order. Indeed,
(nso1) holds since $K^n_n$ is closed under cyclic permutations,
(nso2) is true as $K^n_n$ forbids odd permutations of tuples with
pairwise distinct coordinates and fixed first coordinate, (nso3)
is satisfied since an additional element $t$ either repeats some
coordinate of forms an even permutation of $(A_1,A_2,$ $\ldots,
A_n)$ with respect to given elements $x_1,x_2,\ldots,x_n$, and
(nso4) holds since for any pairwise distinct elements
$x_1,x_2,\ldots,x_n$ either $(x_1,x_2\ldots,x_n)$ is an even
per\-mu\-ta\-t\-i\-on of $(A_1,A_2,\ldots,$ $A_n)$ or an arbitrary
transposition for $(x_1,x_2\ldots,x_n)$ is an even permutation of
$(A_1,A_2,\ldots, A_n)$.

By the definition the relations $K^n_n$ are generated from the
tuple $(A_1,A_2,\ldots,A_n)$ by its even permutations with fixed
$A_1$, cyclic permutations and identifications of coordinates.

Similarly to $K^4_4$ we obtain
\begin{equation}\label{eq_knn}\vert K^n_n \vert
=n^n-\frac{n!}{2}.
\end{equation}
Indeed, by the definition $\vert I_n\vert = n^n - n!$. Now we
calculate the number of possible permutations of $(A_1,
A_2,\ldots, A_n)$. Since we fix $A_1$ there are $\frac{(n-1)!}{2}$
of even per\-mu\-ta\-t\-i\-ons. As we consider the closure by the
permutation $(1,2,\ldots,n)$, we multiply $\frac{(n-1)!}{2}$ by
$n$ obtaining $\frac{n!}{2}$ possibilities. Thus, $\vert K^n_n
\vert=n^n - n! + \frac{n!}{2} = n^n -\frac{n!}{2}$.

The formula (\ref{eq_knn}) is valid for $2$-element linear orders
and $3$-element circular orders. Thus we have $\vert K^2_2\vert
=3$, $\vert K^3_3\vert =24$, $\vert K^4_4\vert =244$, $\vert
K^5_5\vert =3065$, etc.

For instance, $K^5_5$ is generated by twelve directed pentahedrons
$$(A_1,A_2,A_3,A_4,A_5), (A_1,A_2,A_4,A_5,A_3), (A_1,A_2,A_5,A_3,A_4),$$

$$(A_1,A_3,A_2,A_5,A_4), (A_1,A_3,A_4,A_2,A_5), (A_1,A_3,A_5,A_4,A_2).$$

$$(A_1,A_4,A_2,A_3,A_5), (A_1,A_4,A_3,A_5,A_2), (A_1,A_4,A_5,A_2,A_3),$$

$$(A_1,A_5,A_2,A_4,A_3), (A_1,A_5,A_3,A_2,A_4), (A_1,A_5,A_4,A_3,A_2)$$
using cyclic permutations and $I_5$.}
\end{example}

\begin{note}\label{rem131} {\rm Constructing $n$-spherical orders
on $n$-element sets we start to use even permutations since $n=4$
because for $n=2$ there are no nonidentical even permutations and
for $n=3$ these permutations are reduced to cyclic permutations.}
\end{note}

We denote by $S_n:=S(A_1,A_2,\ldots,A_n)$ the unique
$(n-1)$-dimensional sphere containing the points
$A_1,A_2,\ldots,A_n$ in Example \ref{ex13}, and by
$H_n:=H(A_1,A_2,\ldots,A_n)$ the union of hyperplanes containing
$(n-1)$-element subsets of $\{A_1,A_2,\ldots,A_n\}$.

\begin{example}\label{ex14} {\rm
Any relation $K^n_n$ can be extended till an $n$-spherical order
$K^{n+k}_n$ by adding new points $A_{n+1},\ldots,A_{n+k}\in
S_n\setminus H_n$, where the relation $K_n^{n+k}$ is defined as
follows. We choose arbitrary $n$ elements $A_{j_1}, \dots,
A_{j_n}$ from $\{ A_1, \dots, A_{n+k}\}$ whe\-re $j_1 < j_2 < \dots
< j_n$ and consider all even permutations $\tau$ of $\{ 2, 3,
\dots, n\}$. 

Let $\sigma = (1, 2, 3, \dots, n)$ and
$\rho=\sigma^i\tau$ for some $i<n$. Then $K_n^{n+k}$ contains
$(A_{j_{\rho(1)}}, \dots, A_{j_{\rho(n)}}) \in K_n^{n+k}$ as well
as all tuples with at least two equal coordinates. Thus we define
the structures
$\langle\{A_1,\ldots,A_n,A_{n+1},\ldots,A_{n+k}\},K^{n+k}_n\rangle$
generalizing the structures with the relations $K^n_n$ in Example
\ref{ex13}. Since there are $A_{n+k}^n=\frac{(n+k)!}{k!}$
$n$-tuples with distinct coordinates, by simple combinatorial
calculation, we obtain
$$\vert K^{n+k}_n\vert =(n+k)^n-\frac{A_{n+k}^n}{2}.$$

In particular, we have 

$\vert K^3_2 \vert =3^2-\frac{3!}{2}=6$,
$\vert K^4_2\vert =4^2-\frac{4!}{4}=10$, $\vert K^5_2 \vert
=5^2-\frac{5!}{12}=15$, etc.; 

$\vert K^4_3\vert
=4^3-\frac{4!}{2}=52$, $\vert K^5_3\vert =5^3-\frac{5!}{4}=95$,
$\vert K^6_3\vert =6^3-\frac{6!}{12}=156$, etc.;

 $\vert K^5_4\vert
=5^4-\frac{5!}{2}=565$, $\vert K^6_4\vert =6^4-\frac{6!}{4}=1116$,
$\vert K^7_4\vert =7^4-\frac{7!}{12}=1981$, etc. }
\end{example}

\begin{note}\label{rem141} {\rm The construction of finite $n$-spherical orders
shows that these orders of given finite cardinality are unique up
to isomorphism.}
\end{note}

\begin{note}\label{rem15}
{\rm The process of adding $B_1,\ldots,B_k$ in Example \ref{ex14}
can be continued arbitrarily many (unlimit and limit) steps
obtaining $\lambda$-element $n$-spherical orders $K^\lambda_n$ on
$S_n$, for $\lambda$ with $n\leq\lambda\leq 2^\omega$, and on
appropriate elementary extensions $S^\lambda_n$ of $S_n$, for
$\lambda> 2^\omega$.

In particular, taking a sphere $S$ in $\mathbb R^3$ we can
interpret a $4$-spherical order taking the set of all $4$-tuples
$(A_1,A_2,A_3,A_4)$, where $A_4$ is situated inside the spherical
triangle $A_1A_2A_3$ \cite{Stepanov}. It can be naturally spread
for group trigonometries based on ordered groups \cite{SuGP}.

All these orders on spheres are called {\em spherical models}.

The definition of $K^\lambda_n$ implies that this relation cannot
be reduced to Boolean combinations of relations with less arities,
then the arity of the formula $K^\lambda_n(x_1,\ldots,$ $x_n)$
equals $n$: ${\rm ar}(K^\lambda_n(x_1,\ldots,x_n))=n$.

The formula witnessing the identification of coordinates for the
formula $K_n(x_1,$ $\ldots,x_n)$ of $n$-spherical order will be
denoted by $I_n(x_1,\ldots,x_n)$. Clearly, $${\rm
ar}(I_n(x_1,\ldots,x_n))=1,$$ since $I_n(x_1,\ldots,x_n)$ is
equivalent to the formula $$\bigvee\limits_{1\leq i<j\leq n}
\left(x_i\approx
x_j\wedge\bigwedge\limits_{k\notin\{i,j\}}x_k\approx x_k\right).$$
Thus, ${\rm ar}(K_n(x_1,\ldots,x_n)\wedge\neg
I_n(x_1,\ldots,x_n))=n$.}
\end{note}

\section{Countably categorical $n$-spherical orders}\label{sec4}

%\medskip
{\bf Definition.} An $n$-spherically ordered set $\langle A,
K_n\rangle$, where $n\geq 2$, is called {\em dense} if it contains
at least two elements and for each $(a_1,a_2,a_3,\ldots,a_n)\in
K_n$ with $a_1\ne a_2$ there is $b\in A\setminus
\{a_1,a_2,\ldots,a_n\}$ such that
$$
\models K_n(a_1,b,a_3,\ldots,a_n)\wedge K_n(b,a_2,a_3,\ldots,a_n).
$$

Clearly, dense $n$-spherical orders $K_n$ witness the strict order
property producing unstable structures $\langle A,K_n\rangle$,
since fixing $n-2$ distinct coordinates in the relation $K_n$ we
obtain a dense linear order.

The following theorem generalizes the known result on countable
categoricity of dense linear orders \cite[Proposition
3.1.7]{ErPa}.

\begin{theorem}\label{th_dense_isom}
If $\mathcal{A}$ and $\mathcal{B}$ are countable dense
$n$-spherical orders, $n\geq 2$, without endpoints for $n=2$, then
$\mathcal{A}\simeq\mathcal{B}$.
\end{theorem}

Proof. The case $n=2$ is considered in \cite[Proposition
3.1.7]{ErPa}. Therefore we assume that $n\geq 3$, and for this
case we slightly modify the arguments for that proposition.

Let $A=\{a_m\mid m\in\omega\}$, $B=\{b_m\mid m\in\omega\}$. We
consider the set $G$ consisting of maps $g\mbox{\rm : }
A_1\rightarrow B_1$ satisfying the following conditions:

1) $A_1$ and $B_1$ are finite subsets of $A$ and $B$,
respectively;

2) $g\mbox{\rm : }\mathcal{A}(A_1)\iso\mathcal{B}(B_1)$ if
$A_1\neq\emptyset$, i.e., $g$ is an isomorphism between a
substructure $\mathcal{A}(A_1)$ of $\mathcal{A}$ with the universe
$A_1$ and a substructure $\mathcal{B}(B_1)$ of $\mathcal{B}$ with
the universe $B_1$;

3) if $\vert A_1\vert =2m>0$ then $\{a_0,\dots,a_{m-1}\}\subseteq
A_1$ and $\{b_0,\dots,b_{m-1}\}\subseteq B_1$;

4) if $\vert A_1 \vert =2m+1$ then $\{a_0,\dots,a_m\}\subseteq
A_1$ and for $m>0$, $\{b_0,\dots,b_{m-1}\}\subseteq B_1$.

Since $\emptyset\in G$ then $G\neq\emptyset$. Let $g\mbox{\rm : }
A_1\rightarrow B_1$ belong to $G$ and $\vert A_1\vert =2n$. In
view of 3) there is $a\in A\setminus A_1$ with
$\{a_0,\dots,a_m\}\subseteq A_1\cup\{a\}$. Now we find an element
$b\in B\setminus B_1$ such that
$K_n(b,g(c_2),\ldots,g(c_n))\Leftrightarrow K_n(a,c_2,\ldots,c_n)$
for all $c_2,\ldots,c_n\in A_1$. The element $b$ exists since
$\mathcal{B}$ is dense and the condition 2) for $g$ holds.

Clearly, $g\cup\{\langle a,b \rangle\}\in G$. If $\vert A_1\vert
=2n+1$ then we replace $\mathcal{A}$ and $\mathcal{B}$ each other
and find a pair $\langle a,b \rangle\notin g$ with $g\cup\{\langle
a,b \rangle\}\in G$. Thus the partial order $\langle
G,\subseteq\rangle$ does not have maximal elements. Hence $G$
contains an infinite chain $X\subseteq G$. In view of 2)--4) the
union $\bigcup X$ is an isomorphism of $\mathcal{A}$ onto
$\mathcal{B}$. \endproof

\medskip
Theorem \ref{th_dense_isom} immediately implies:

\begin{corollary}\label{cor_dcc}
For any dense $n$-spherical order $\mathcal{A}$ its theory ${\rm
Th}(\mathcal{A})$ is countably categorical.
\end{corollary}

\begin{note}\label{rem_ccso}
{\rm Extending dense $n$-spherical orders by finitely many new
discrete elements preserves countable categoricity. It means that
there are finitely many possibilities for
$(a_1,a_2,a_3,\ldots,a_n)\in K_n$ such that $a_1\ne a_2$ and there
are finitely many $b\notin\{a_1,a_2\}$ with
$$
\models K_n(a_1,b,a_3,\ldots,a_n)\wedge K_n(b,a_2,a_3,\ldots,a_n).
$$
Similarly linear orders having infinite discrete parts loose the
countable categoricity obtaining a definable successor function.}
\end{note}

Similarly the theories of dense linear orders, in view of Lemma
\ref{lem01}, the theories $T_n$ of dense $n$-spherical orders
$\langle A, K_n\rangle$, $n\geq 2$, admit the quantifier
elimination since complete types are forced by collections of
quantifier free formulae. Besides the theories $T_n$ are finitely
axiomatizable. Using Corollary \ref{cor_dcc} and the arguments for
\cite[Proposition 8.3.1]{ErPa} we obtain the following its
generalization:

\begin{theorem}\label{th_dec}
For any natural $n\geq 2$ the theory $T_n$ of a dense
$n$-spherical order is decidable.
\end{theorem}

\section{Ehrenfeucht theories based on dense $n$-sphe\-ri\-cal orders}\label{sec5}

In this section we modify classical Ehrenfeucht examples on linear
dense orders \cite{CCMCT, Va} to Ehrenfeucht theories based on
dense $n$-sphe\-ri\-cal orders and producing arbitrarily finitely
many countable models.

For a theory $T$, we denote by
$I(T,\lambda)$\index{$I(T,\lambda)$} the number of pairwise
non-isomorphic models of $T$ in~a~power~$\lambda$. The value
$I(T,\lambda)$ is called the {\em $\lambda$-spectrum} of $T$. If
$\lambda=\omega$ then the $\lambda$-spectrum of $T$ is called the
{\em countable spectrum} of $T$.

\medskip
{\bf Definition} \cite{Mi2}. A theory $T$ is \emph{Ehrenfeucht} if
$1<I(T,\omega)<\omega,$ i.e., the countable spectrum of $T$ is
finite and greater than $1$.

\medskip
Let $\mathcal{A}_n=\langle A_n, K_n\rangle$ be a countable dense
$n$-spherical order, $n\geq 2$. Let $T^m_n$ be the theory of a
structure $\mathcal{M}^m_n$, formed from the structure
$\mathcal{A}_n$ by adding pairwise distinct constants $c_k$,
$k\in\omega$, such that $\models K_n(c_{i_1},\ldots,c_{i_n})$ for
any $i_1<\ldots<i_n$, and unary predicates $P_0,\ldots,P_{m-3}$
which form a~partition of the set $A_n$, with
$$\models\forall x_1,\ldots,x_n\:\Biggm(\bigwedge\limits_{1\leq j<k\leq n}\neg(x_j\approx x_k)\wedge K_n(x_1,\ldots,x_n)\to$$
$$\to\exists t\:\Biggm(\bigwedge\limits_{j=1}^n\neg(t\approx
x_j)\wedge K_n(x_1,t,x_3,\ldots,x_n)\wedge
K_n(t,x_2,x_3,\ldots,x_n)\wedge P_i(t)\Biggm)\Biggm),$$
$i=0,\ldots,m-3.$

We denote by $p_\infty(x)$ the type which is forced by the set of
formulae $$K_n(c_{i_1},c_{i_2},\ldots,c_{i_{n-1}},x),$$
$i_1<i_2<\ldots<i_{n-1}$. The type $p_\infty(x)$ has $m-2$
completions $p^i_\infty(x)$ which are forced by
$p_\infty(x)\cup\{P_i(x)\}$, $i=0,\ldots,m-3.$

The theory $T^m_n$ has exactly $m$ pairwise non-isomorphic
countable models:

(a) a prime model $\mathcal{M}_m$ which omit the type
$p_\infty(x)$; ($\lim\limits_{k\to\infty}c_k=\infty$);

(b) prime models $\mathcal{M}^i_m$ over realizations of types
$p^i_\infty(x)$, $i=0,\ldots,m-3$, with a limit element for the
constants $c_k$, realizing $p^i_\infty(x)$;

(c) a saturated model $\overline{\mathcal{M}}_m$, without limit
elements for the constants $c_k$, realizing $p^i_\infty(x)$.

Replacing constants $c_k$ by unary predicates $U_k=\{c_k\}$ we
obtain theories $\widehat{T}^m_n$ instead of $T^m_n$ such that
$\widehat{T}^m_n$ are unary expansions of $T_n={\rm
Th}(\mathcal{A}_n)$, i.e. expansions of $T_n$ by unary predicates.
It is easy to see that this transformation preserves the number of
pairwise non-isomorphic countable models:
$$
I(\widehat{T}^m_n,\omega)=m=I(T^m_n,\omega).
$$

Corollary \ref{cor_dcc} and the arguments above producing values
for $I(\widehat{T}^m_n,\omega)$ imply the following:

\begin{theorem}\label{th_dso_spec}
For any $m\in\omega\setminus\{0,2\}$, $n\in\omega\setminus\{0,1\}$
there is a unary expansion $\widehat{T}^m_n$ of the theory $T_n$
of dense $n$-spherical order $\mathcal{A}_n$ such that $
I(\widehat{T}^m_n,\omega)=m$.
\end{theorem}

The theories $T^m_2$ in Theorem \ref{th_dso_spec} are classical
Ehrenfeucht examples, and the theories $T^m_3$ are their
adaptations for dense circular orders.

Figure \ref{fig222a} illustrates a model of the theory $T^3_4$.
Constants $c_k$, $k\in\omega$, are situated on a spiral with a top
limit vertex $\ast$.


\begin{figure}
\begin{center}
\setlength{\unitlength}{1.1mm}
\begin{picture}(45,45)(4.5,11)

\qbezier(45.0, 30.0)(45.0, 38.2843) (39.1421, 44.1421)
\qbezier(39.1421, 44.1421)(33.2843, 50.0) (25.0, 50.0)
\qbezier(25.0, 50.0)(16.7157, 50.0) (10.8579, 44.1421)
\qbezier(10.8579, 44.1421)(5.0, 38.2843) (5.0, 30.0) \qbezier(5.0,
30.0)(5.0, 21.7157) (10.8579, 15.8579) \qbezier(10.8579,
15.8579)(16.7157, 10.0) (25.0, 10.0) \qbezier(25.0, 10.0)(33.2843,
10.0) (39.1421, 15.8579) \qbezier(39.1421, 15.8579)(45.0, 21.7157)
(45.0, 30.0)



\put(25,50){\makebox(0,0)[cc]{$\ast$}}
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\put(20.5,13.2){\makebox(0,0)[cc]{$c_0$}}
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\put(41,22){\makebox(0,0)[cc]{$c_1$}}
\put(41,39){\makebox(0,0)[cc]{$\bullet$}}
\put(40.9,37){\makebox(0,0)[cc]{$c_2$}}

\put(18,42.6){\makebox(0,0)[cc]{$\bullet$}}
\put(18,45){\makebox(0,0)[cc]{$c_3$}}

\qbezier(5,30)(25,18)(45,30) %\qbezier(5,35)(25,45)(45,35)

\qbezier(5,30)(6.5,30.95)(8,31.45)
\qbezier(10,32.2)(11.5,32.85)(13,33.2)
\qbezier(15,33.8)(16.5,34.2)(18,34.5)
\qbezier(20,34.9)(23,35.4)(24,35.3)
\qbezier(26,35.3)(27,35.4)(30,34.9)
\qbezier(32,34.5)(33.5,34.2)(35,33.8)
\qbezier(37,33.2)(38.5,32.85)(40,32.2)
\qbezier(42,31.45)(43.5,30.95)(45,30)

\qbezier(25,10)(13,30)(25,50)

\qbezier(25,10)(25.95,11.5)(26.45,13)
\qbezier(27.2,15)(28.0,17.5)(28.15,18)
\qbezier(28.7,20)(29.1,21.5)(29.4,23)
\qbezier(29.8,25)(30.3,28)(30.2,29)
\qbezier(30.2,31)(30.3,34)(29.8,35)
\qbezier(29.4,37)(29.1,38.5)(28.7,40)
\qbezier(28.15,42)(27.8,43.5)(27.2,45)
\qbezier(26.45,47)(25.95,48.5)(25,50)

\qbezier(15,15)(43,16)(44.1,35.8)
\qbezier(44.1,35.8)(43,39)(38,40.3)
\qbezier(36,40.8)(33,41.7)(30,42)
\qbezier(28,42.3)(25,42.8)(22,42.7)
\qbezier(20,42.7)(17,42.8)(14,42.6)
\qbezier(12,42.51)(11.5,42.55)(9.2,42.29)
\qbezier(9.2,42.29)(36.9,37.55)(34.5,47.5)

\put(32,47.6){\makebox(0,0)[cc]{$\ldots$}}

\end{picture}
\end{center}
\caption{} \label{fig222a}
\end{figure}

\section{Constant expansions of dense $n$-sphe\-ri\-cal orders and their countable spectra}\label{sec6}

Now we consider possibilities for distributions of countable
sequences of constants expanding $n$-spherical theories $T_n={\rm
Th}(\mathcal{A}_n)$, where $\mathcal{A}_n=\langle A_n,K_n\rangle$
are dense $n$-spherical orders, $n\geq 2$. These distributions
produce distributions of countable models of these expansions
$T\supseteq T_n$ and possibilities for values $I(T,\omega)$.

The distributions for $n=2$ are described in \cite{CCMCT, Ma}.

Recall \cite{PilSt} that a linearly ordered structure
$\mathcal{M}$ is {\em $o$-minimal}\index{Structure!$o$-minimal} if
any definable (with parameters) subset of $M$ is a finite union of
singletons and open intervals $(a,b)$, where $a\in
M\cup\{-\infty\}$, $b\in M\cup\{+\infty\}$. A theory $T$ is {\em
$o$-minimal}\index{Theory!$o$-minimal} if each model of $T$ is
$o$-minimal.

As examples of Ehrenfeucht $o$-minimal theories, we mention the
theories $T^1\rightleftharpoons{\rm Th}(({\mathbb
Q};<,c_n)_{n\in\omega})$ and $T^2\rightleftharpoons{\rm
Th}(({\mathbb Q};<,c_n,c'_n)_{n\in\omega})$, where $<$ is an
ordinary strict order on the set ${\mathbb Q}$ of rationals, the
constants $c_n$ form a strictly increasing sequence, and the
constants $c'_n$ form a strictly decreasing sequence, $c_n<c'_n$,
$n\in\omega$.

The theory $T^1$ is an Ehrenfeucht's example \cite{Va} with
$I(T^1,\omega)=3$. It has two almost prime models and one limit
model:

{\small $\bullet$} a prime model with the empty set of
realizations of the type $p(x)$ isolated by the set $\{c_n<x\mid
n\in\omega\}$ of formulae;

{\small $\bullet$} a prime model over a realization of the type
$p(x)$, with the least realization of that type;

{\small $\bullet$} one limit model over the type $p(x)$, with the
set of realizations of $p(x)$ forming an open convex set.

The Hasse diagram for the Rudin--Keisler preorder $\leq_{\rm RK}$
and values of the func\-tion ${\rm IL}$ of distributions of numbers
of limit models for $\sim_{\rm RK}$-classes of $T^1$ is
re\-pre\-sen\-ted in Figure~\ref{fig3}. This Hasse diagram equals the
Hasse diagram for the theory $T^3_4$.\footnote{Hasse diagrams for
distributions of countable models are defined and studied for the
general case \cite{CCMCT}, for quite $o$-minimal Ehrenfeucht
theories \cite{KulSudRK}, and for disjoint unions of Ehrenfeucht
theories \cite{RKDU}.}

\begin{figure}[t]
\begin{center}
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\end{center}
\hspace{-45mm}\parbox[t]{0.3\textwidth}{\caption{}\label{fig3}}
\hfill
\hspace{-86mm}\parbox[t]{0.47\textwidth}{\caption{}\label{fig4}}
\hfill
\hspace{-104mm}\parbox[t]{0.65\textwidth}{\caption{}\label{fig5}}

\end{figure}

The theory $T^2$ has the six pairwise non-isomorphic countable
models:

{\small $\bullet$} a prime model with the empty set of
realizations of the type $p(x)$ isolated by the set $\{c_n<x\mid
n\in\omega\}\cup\{x<c'_n\mid n\in\omega\}$;

{\small $\bullet$} a prime model over a realization of $p(x)$,
with a unique realization of this type;

{\small $\bullet$} a prime model over a realization of the type
$q(x,y)$ isolated by the set $p(x)\cup p(y)\cup\{x<y\}$; here the
set of realizations of $p(x)$ forms a closed interval $[a,b]$;

{\small $\bullet$} three limit models over the type $q(x,y)$, in
which the sets of realizations of $q(x,y)$ are convex sets of
forms $(a,b]$, $[a,b)$, $(a,b)$ respectively.

In Figure \ref{fig4} we represent the Hasse diagram
of~Rudin--Keisler preorders $\leq_{\rm RK}$ and values of
distribution functions ${\rm IL}$ of numbers of limit models on
$\sim_{\rm RK}$-equivalence classes for the theory $T^2$.

The following theorem shows that the number of countable models of
Ehrenfeucht $o$-minimal theories is exhausted by combinations of
these numbers for the theories $T^1$ and $T^2$.

\begin{theorem}\label{th1Ma} {\rm \cite{Ma}}
Let $T$ be an $o$-minimal theory in a countable language. Then
either $T$ has $2^{\omega}$ countable models or $T$ has exactly
$3^r\cdot 6^s$ countable models, where $r$ and $s$ are natural
numbers. Moreover, for any $r,s\in\omega$ there is an $o$-minimal
theory $T$ with exactly $3^r\cdot 6^s$ countable models.
\end{theorem}

Theorem \ref{th1Ma} immediately implies:

\begin{corollary}\label{co2Ma}
Let $T$ be a countable constant expansion of the $2$-spherical
theory $T_2$. Then either $T$ has $2^{\omega}$ countable models or
$T$ has exactly $3^r\cdot 6^s$ countable models, where $r$ and $s$
are natural numbers. Moreover, for any $r,s\in\omega$ there is an
aforesaid theory $T$ with exactly $3^r\cdot 6^s$ countable models.
\end{corollary}

Since any $3$-spherical structure is obtained by placing a linear
order on a circle, the arguments for the $2$-spherical theory
$T_2$ are valid for the $3$-spherical theory $T_3$ obtaining the
following:

\begin{corollary}\label{co3Ma}
Let $T$ be a countable constant expansion of the $3$-spherical
theory $T_3$. Then either $T$ has $2^{\omega}$ countable models or
$T$ has exactly $3^r\cdot 6^s$ countable models, where $r$ and $s$
are natural numbers. Moreover, for any $r,s\in\omega$ there is an
aforesaid theory $T$ with exactly $3^r\cdot 6^s$ countable models.
\end{corollary}

Considering the dense $n$-spherical Ehrenfeucht theories $T_n$,
$n\geq 4$, we obtain both the possibilities $3^k\cdot 6^s$ of
countable models, with appropriate sequences of constants, and the
following new possibilities.

Considering a consistent non-isolated set $p(x)$ of formulae
$$K^\delta_n(c_{j_1},\ldots,c_{j_{i-1}},x,c_{j_{i+1}},\ldots,c_{j_n})\wedge\bigwedge\limits_{k\ne
i}\neg x\approx c_{j_k},$$
$\delta=\delta_{j_1,\ldots,j_{i-1},x,j_{i+1},\ldots,j_n}\in\{0,1\}$,
with pairwise distinct
$c_{j_1},\ldots,c_{j_{i-1}},c_{j_{i+1}},\ldots,c_{j_n}$, we
observe that each such an additional formula divides the domain of
previous ones into two parts with respect to a $(n-1)$-dimensional
plane containing points
$c_{j_1},\ldots,c_{j_{i-1}},c_{j_{i+1}},\ldots,c_{j_n}$. In view
of quantifier elimination for $T_n$ any consistent set $p(x)$
forces a complete type. Thus a limit part of the domain being a
set of solutions for $p(x)$ is defined by the limits for
$c_{j_1},\ldots,c_{j_{i-1}},c_{j_{i+1}},\ldots,c_{j_n}$.

We have the following possibilities for these limits:

{\small $\bullet$} $p(x)$ is omitted;

{\small $\bullet$} $p(x)$ has a unique realization $a_\infty$
being a common limit of the sequences;

{\small $\bullet$} $p(x)$ has infinitely many realizations
independently including / not including $n-1$ limits with respect
to coordinates of
$(c_{j_1},\ldots,c_{j_{i-1}},c_{j_{i+1}},\ldots,c_{j_n})$.

Hence there are $2^{n-1}+2$ possibilities for countable models,
where 3 models are prime over finite sets (prime over $\emptyset$,
prime over $\{a_\infty\}$, and prime over limits for
$c_{j_1},\ldots,c_{j_{i-1}},c_{j_{i+1}},\ldots,c_{j_n}$) and
$2^{n-1}-1$ limit models.

If some constants in
$c_{j_1},\ldots,c_{j_{i-1}},c_{j_{i+1}},\ldots,c_{j_n}$  are
fixed, with $k>1$ independent moving sequences the total number of
countable models related to $p(x)$ equals $2^k+2$ including $3$
almost prime models and $2^k-1$ limit models, see Figure
\ref{fig5}.

Since distinct types $p(x)$ are independent the total number of
possibilities for countable models is obtained by the
multiplications of values $2^k+2$ for various $k\in
n\setminus\{1\}$, if there are finitely many non-isolated
$1$-types, and there are $2^{\omega}$ countable models otherwise.
Thus we have the following:

\begin{theorem}\label{thsphEr}
Let $T$ be a countable constant expansion of the $n$-spherical
theory $T_n$, $n\geq 3$. Then either $T$ has $2^{\omega}$
countable models or $T$ has exactly $\prod\limits_{k\in
n\setminus\{1\}}(2^k+2)^{r_k}$ countable models, where $r_k$ are
natural numbers. Moreover, for any $r_0,\ldots,r_{n-1}\in\omega$
there is an aforesaid theory $T$ with exactly $\prod\limits_{k\in
n\setminus\{1\}}(2^k+2)^{r_k}$ countable models.
\end{theorem}

Theorem \ref{thsphEr} confirms the Vaught conjecture for countable
constant expansions $T$ of $n$-spherical theories $T_n$. In
particular, $$\mbox{ either }I(T,\omega)=2^\omega\mbox{ or }
I(T,\omega)=3^{r_1}\cdot 6^{r_2}\cdot 10^{r_3}$$ for $T\supseteq
T_4$, $$\mbox{ either }I(T,\omega)=2^\omega\mbox{ or }
I(T,\omega)=3^{r_1}\cdot 6^{r_2}\cdot 10^{r_3}\cdot 18^{r_4}$$ for
$T\supseteq T_5$, etc.

Similarly to the quite $o$-minimal Ehrenfeucht theories
distributions of countable models of constant expansions of
$n$-spherical theories are given by Hasse diagrams for disjoint
unions of theories \cite{KulSudRK, RKDU}, based on the diagrams
represented in Figures \ref{fig3}--\ref{fig5}.

\section{Conclusion}\label{sec7}

We studied spherical orders, which generalize known linear and
circular orders. Semantic and syntactic properties of spherical
orders and their elementary theories, including finite and dense
orders and their theories are investigated. It is shown that
theories of dense $n$-spherical orders are countably categorical
and decidable. The values for spectra of countable models of unary
expansions of $n$-spherical theories are described. The Vaught
conjecture is confirmed for countable constant expansions of dense
$n$-spherical theories. It would be natural to study various
modifications and kinds of spherical orders and their theories.

\medskip
{\bf Acknowledgement}. The authors thank the anonymous referee for
the co\-n\-struc\-ti\-ve critics and suggestions contributing to
the improvement of presentation.

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\end{document}
