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\title[Minimality conditions]{Minimality conditions, topologies, and ranks \\ for spherically ordered 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.,
Minimality conditions, topologies, and ranks for spherically
ordered theories}
\thanks{\copyright \ 2023 Sudoplatov S.V}
\thanks{\rm The work
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, ??, 2022, published  ???, ??,  2023.}%


\semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} The class of ordered structures is
productively studied both in order to classify them and in various
applications connected with comparing of objects and information
structuring. Important particular kinds of ordered structures are
represented by $o$-minimal, weakly $o$-minimal and circularly
minimal ones as well as their variations including definable
minimality. We show that the well developed powerful theory for
$o$-minimality, circular minimality, and definable minimality is
na\-tu\-ral\-ly spread for the spherical case. Reductions of
spherical orders to linear ones, called the linearizations, and
back reconstructions, called the sphe\-ri\-fi\-ca\-t\-ions, are
examined. Neighbourhoods for spherically ordered struc\-tu\-res
and their topologies are studied. It is proved that related
topological spaces can be $T_0$-spaces, $T_1$-spaces and Hausdorff
ones. These cases are characterized by the cardinality estimates
of the universe. Definably minimal linear orders, their definably
minimal extensions and restrictions as well as spherical ones are
described. The notion of convexity rank is generalized for
spherically ordered theories, and values for the convexity rank
are realized in weakly spherically minimal theories which are
co\-un\-t\-ab\-ly categorical.\medskip

\noindent{\bf Keywords:} spherical order, weak spherical
minimality, definable minimality, topology, convexity rank.
 \end{quote}
}

\bigskip
\section{Introduction}

Ordered structures and their theories are broadly used in
Mathematics and applications both for studying various structural
aspects of reality and for ranking objects of different nature in
order to compare and structure them. There are many classification
results for partially \cite{Fuchs, Glass, DP, Schr}, linearly
\cite{Ros}, circularly \cite{Nov, KulMac} ordered structures,
their elementary theories \cite{KulMac, Ta11} and automorphism
groups \cite{DGM, Truss}. There is a deep theory classifying
$o$-minimal structures \cite{PiSt, VDD}, weakly $o$-minimal
structures \cite{MMS} and their variations \cite{MacSt, Tof, Wen}.

Recall \cite{PiSt} that a linearly ordered structure
$\mathcal{M}=\langle M,=,<,\ldots \rangle$ is said to be {\em
$o$-minimal} if any definable (with parameters) subset of
$\mathcal{M}$ is the union of finitely many intervals and points
in~$\mathcal{M}$. We also recall \cite{MMS} that such a structure
$\mathcal{M}$ is {\it weakly $o$-minimal} if any definable (with
parameters) subset of $M$ is a finite union of convex sets
in~$\mathcal{M}$.

In the present paper we propose an approach classifying spherical
generalizations of $o$-minimal and weakly $o$-minimal theories and
structures via minimal conditions studied in \cite{KulMac, Ta11,
BMN} and of related topologies \cite{Engel} connected with tame
ones \cite{VDD}.

The paper is organized as follows. Preliminary notions on circular
and spherical orders, and topological terminology are represented
in Section~2. In Section 3, notions on spherical minimality are
introduced, links between linear and spherical orders are
described. Topological properties related to spherical orders are
studied in Section 4. In Section 5, we describe links between
linear and spherical definable minimalities. Convexity rank and
its realizations for spherically ordered theories are described in
Section 6.

\section{Preliminaries}

We denote by $\mathcal{A},\mathcal{B},\mathcal{C},\ldots$
structures and by $A,B,C,\ldots$ correspondent universes of these
structures. By ${\rm Th}(\mathcal{A})$ we denote the complete
first-order theory of the structure $\mathcal{A}$. Throughout we
use standard model-theoretic \cite{Sh, Hodges, TZ} and topological
\cite{Engel} notions and notations.

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) $\forall x\forall y \forall z(K_3(x,y,z)\to \forall
 t[K_3(x,y,t)\lor
 K_3(x,t,z)\lor K_3(t,y,z)]);$\\
 (co4) $\forall x\forall y \forall z (K_3(x,y,z)\lor K_3(y,x,z)).$

\medskip
In fact circular orders are obtained from linear ones by their
representations on circles.

\medskip
Following \cite{aafot} 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 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 ${\rm ar}_T(\varphi)$.

Clearly, ${\rm ar}(K_3(x,y,z))=3$ if the relation has at least
three element domain, i.e., $K_3(x,y,z)$ is not reduced to Boolean
combinations of formulae with at most two free variables. Hence,
theories with infinite circular order relations are at least
$3$-ary.

The following generalization of a circular order produces a {\em
$n$-ball}, or {\em $n$-spherical}, or {\em $n$-circular} order
relation \cite{aafot, ananat, KS4}, for $n\geq 3$, which is
described by a $n$-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) $\forall x_1,\ldots,x_n (K_n(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)\lor$
 $$\lor K_n(x_1,\ldots,x_j,\ldots,x_i,\ldots,x_n)),\,\, 1\leq i<j\leq
 n.$$

\medskip
Clearly, the axioms for $n$-spherical orders can be naturally
adapted for $2$-spherical, i.e., linear ones producing a linear
order $K_2$. Here (nso2) gives 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) give 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.

\medskip
Structures $\mathcal{A}=\langle A,K_n\rangle$ with $n$-spherical
orders $K_n$, where the domain $A$ is the set of all coordinates
for tuples in $K_n$, will be called {\em $n$-spherical orderings},
or {\em $n$-spherical orders}, too.

We have ${\rm ar}(K_n(x_1,\ldots,x_n))=n$ for spherical orders
with at least $n$-elements domains \cite{aafot} producing at least
$n$-ary theories with infinite $n$-spherical order relations.

\medskip
{\bf Remark 2.1.} \rm Along with a $n$-spherical order $K_n$ in a
structure $\mathcal{A}=\langle A,K_n\rangle$ a {\em dual}
$n$-spherical order $\overline{K}_n$ is definable which is defined
by the formula
$$
\neg K_n(x_1,\ldots,x_n)\vee \bigvee\limits_{1\leq k<l\leq n}
x_k\approx x_l.
$$
In particular, any linear order $\leq=K_2$ has the dual linear
order $\geq=\overline{K}_2$, any circular order $\leq=K_3$ has the
dual circular order $\overline{K}_3$, etc.

\medskip
{\bf Definition} \cite{Engel}. A {\em topological
space}\index{Space topological} is a pair $(X,\mathcal{O})$
consisting of a set $X$ and a family $\mathcal{O}$ of {\em
open}\index{Subset open} subsets of $X$ satisfying the following
conditions:

(O1) $\varnothing\in\mathcal{O}$ and $X\in\mathcal{O}$;

(O2) If $U_1\in\mathcal{O}$ and $U_2\in\mathcal{O}$ then $U_1\cap
U_2\in\mathcal{O}$;

(O3) If $\mathcal{O}'\subseteq\mathcal{O}$ then
$\cup\mathcal{O}'\in\mathcal{O}$.

\medskip
{\bf Definition} \cite{Engel}. A topological space
$(X,\mathcal{O})$ is a {\em $T_0$-space}\index{$T_0$-space} if for
any pair of distinct elements $x_1,x_2\in X$ there is an open set
$U\in\mathcal{O}$ containing exactly one of these elements.

\medskip
{\bf Definition} \cite{Engel}. A topological space
$(X,\mathcal{O})$ is a {\em $T_1$-space}\index{$T_1$-space} if for
any pair of distinct elements $x_1,x_2\in X$ there is an open set
$U\in\mathcal{O}$ such that $x_1\in U$ and $x_2\notin U$.

\medskip
{\bf Definition} \cite{Engel}. A topological space
$(X,\mathcal{O})$ is a {\em $T_2$-space}\index{$T_2$-space}, or
{\em Hausdorff}\index{Space topological!Hausdorff} if for any pair
of distinct points $x_1,x_2\in X$ there are open sets
$U_1,U_2\in\mathcal{O}$ such that $x_1\in U_1$, $x_2\in U_2$, and
$U_1\cap U_2=\varnothing$.

\section{Linear and spherical orders and their minimalities}

The following assertion connects linear orders and $n$-spherical
orders.

\begin{fact}\label{fact1} {\rm (cf. \cite[Fact 2.2]{KulMac}, \cite[Theorem 11.9]{BMMN})}

{\rm (i)} If $\langle M,\leq\rangle$ is a linear ordering and
$$K_n=\{(a_{i_1},a_{i_2},\ldots
a_{i_n})\mid a_i=a_j\mbox{ for some }i\ne j,$$
$$\mbox{or } a_1<a_2<\ldots <a_n\mbox{ and }(i_1,i_2,\ldots,i_n)\mbox{ is obtained }$$
$$\mbox{by an even permutation of
}(1,2,\ldots,n)\},$$ then $K_n$ is a definable $n$-spherical order
on $M$.

{\rm (ii)} If $\langle M,K_n\rangle$ is a $n$-spherical ordering
and $a\in M$, then the relation $K_{n-1,a}$ defined on
$M_a\rightleftharpoons M\setminus\{a\}$ by the rule
$$
K_{n-1,a}(x_1,\ldots,x_{n-1})\rightleftharpoons
K_n(a,x_1,\ldots,x_{n-1})
$$
is a $(n-1)$-spherical order. Moreover, the relations
$K_{n-m,a_1,\ldots,a_m}$ defined on
$$M_{a_1,\ldots,a_m}\rightleftharpoons
M\setminus\{a_1,\ldots,a_m\},$$ for pairwise distinct
$a_1,\ldots,a_m\in M$, by the rule
$$
K_{n-m,a_1,\ldots,a_m}(x_1,\ldots,x_{n-m})\rightleftharpoons
K_n(a_1,\ldots,a_m,x_1,\ldots,x_{n-m})
$$
are $(n-m)$-spherical orders including the linear order
$K_{2,a_1,\ldots,a_{n-2}}(x_1,x_2)$, where $n-m\geq 2$.
Furthermore, extending this linear order to one, denoted by
$\leq'$ and by $\leq'_{a_1,\ldots,a_{n-2}}$, on $M$ by specifying
that $a_1\leq'\ldots\leq' a_{n-2}\leq' b$ for all $b\in
M_{a_1,\ldots,a_m}$ then the derived $n$-spherical order relation
is the original $n$-spherical order $K_n$.
\end{fact}

{\bf Definition.} The orders
$K_{n-m,a_1,\ldots,a_m}(x_1,\ldots,x_{n-m})$ are called the {\em
$(n-m)$-reductions} of the $n$-spherical order $K_n$, with respect
to $a_1,\ldots,a_m$, and the $2$-reductions
$K_{2,a_1,\ldots,a_{n-2}}(x_1,x_2)$ are called the {\em
linearizations} of $K_n$ (with respect to $a_1,\ldots,a_{n-2}$),
denoted by $\widehat{L}(K_n,a_1,\ldots,a_{n-2})$ or simply
$\widehat{L}(K_n)$.

Thus we obtain the operations reducing $n$-spherical orders to
linear ones. Con\-verse actions transforming linear orders $L$ to
$n$-spherical orders are called the {\em $n$-spherifications}
$\widehat{K}_n(L)$.

\medskip
The following identities hold, with respect to fixed elements
$a_1,\ldots,a_{n-2}$: $$\widehat{L}(\widehat{K}_n(L))=L,
\widehat{K}_n(\widehat{L}(K_n))=K_n.$$

Clearly, the operators $\widehat{L}$ and $\widehat{K}_n$ preserve
the structural complexity, in particular, the
$\omega$-categoricity and rank values for definable sets.

\medskip
{\bf Definition.} Let $A\subseteq M$, where $\mathcal{M}=\langle
M,K_n,\ldots\rangle$ is a $n$-spherically ordered structure. The
set $A$ is called {\em convex} if for any
$a_1,a_2,\ldots,a_{n-1}\in A$ the following holds: for any $b\in
M$ with $\models K_n(a_1,\ldots,a_{n-1},b)$ we have $b\in A$ or
for any $b\in M$ with $\models\neg K_n(a_1,\ldots,a_{n-1},b)$ we
have $b\in A$.

\medskip
{\bf Notation.} 1. For a linearly ordered structure
$\mathcal{M}=\langle M,\leq,\ldots\rangle$ we denote by
$s_n(\mathcal{M})$ the structure $\langle M,K_n,\ldots\rangle$,
where we replace the linear order $\leq$ by a $n$-ary relation
$K_n$, which is derived from $\leq$ following Fact \ref{fact1}(i).
Following \cite{KulMac} the operator $s_3(\cdot)$ is denoted by
$c(\cdot)$.

By Fact \ref{fact1} we also introduce structures
$s^m_n(\mathcal{M})$, for $m\leq n$, where the predicate $K_m$ for
$\mathcal{M}$ is replaced by the predicate $K_n$. In particular,
the operator $s^2_3(\cdot)$ produces a circular order for a given
linear order.

2. Let $\mathcal{M}_0,\mathcal{M}_1,\ldots,\mathcal{M}_{k-1}$ be
linear orderings. Replacing $\leq$ by $K_n$ as above we denote by
$s_n(\mathcal{M}_0+\mathcal{M}_1+\ldots +\mathcal{M}_{k-1})$ the
$n$-spherically ordered sum $\mathcal{M}$ of
$\mathcal{M}_0,\mathcal{M}_1,\ldots,\mathcal{M}_{k-1}$ such that
for each $0\leq i\leq k-1$, considered $({\rm mod}k)$,
$\mathcal{M}_i$ is the immediate predecessor of
$\mathcal{M}_{i+1}$: that is, for $a_1,\ldots,a_j\in M_i$,
$a_{j+2},\ldots,a_n\in M_{i+1}$, and $b\in M$, if $\models
K_n(a_1,\ldots,a_j,b,a_{j+2},\ldots,a_n)$, then $b\in M_i\cup
M_{i+1}$.

3. We put $K^0_n(x_1,\ldots,x_n)\rightleftharpoons
K_n(x_1,\ldots,x_n)\wedge\bigwedge\limits_{i\ne j}\neg x_i\approx
x_j$.

\begin{note} {\rm Notice that the operator $s^2_n$ differs from
$n$-spherification $\widehat{K}_n$ since in the second case $n-2$
new elements are added. At the same time the results of
applications of operators $s^2_n$ and $\widehat{K}_n$ can be
isomorphic or non-isomorphic depending on positions of new
elements $a_1,\ldots,a_{n-2}$.

Indeed, $s^2_3(\omega+\omega^\ast)$ is isomorphic to
$\widehat{K}_3(\omega+\omega^\ast)$ if new element $a_1$ is
situated between elements of $\omega$ or of $\omega^\ast$, and
these structures are not isomorphic if new element $a_1$ is
greater than $\omega$ and less than $\omega^\ast$, obtaining
$\widehat{K}_3(\omega+\omega^\ast)=c(\omega+1+\omega^\ast)$. }
\end{note}

\medskip
{\bf Definition.} Let $S\subseteq M$, where $\mathcal{M}=\langle
M,K_n,\ldots\rangle$ is a $n$-spherically ordered structure. The
set $S$ is said to be an {\em open $n$-segment surface} if $S
=\{b\in M \mid \mathcal{M}\models K^0_n(a_1,b,a_3,\ldots,a_n)\}$
for some pairwise distinct $a_1,a_3,\ldots,a_n\in M$. It has
endpoints $a_1,a_3,\ldots,a_n$. For an open $n$-segment surface
$S$ of a $n$-spherically ordered set, sometimes we will write
$S=(a_1,a_3,\ldots,a_n)$ if we wish to indicate the endpoint frame
of $S$. Similarly, we may define {\em closed}, {\em partially
open}, etc., $n$-segment surfaces in $\mathcal{M}$ including
all/some coordinates $a_i$. By a {\em $n$-segment surface} in
$\mathcal{M}$ we shall mean, ambiguously, any of the above types
of $n$-segment surfaces in $\mathcal{M}$.

\medskip
It is obvious that both a $n$-segment surface and a point are
convex sets.

\medskip
{\bf Definition.} Let $\mathcal{M}=\langle M,K_n,\ldots\rangle$ be
a $n$-spherically ordered structure. The structure $\mathcal{M}$
is said to be {\em $n$-spherically minimal} if any definable (with
parameters) subset of $M$ is a positive Boolean combination of
segment surfaces and points in $\mathcal{M}$. The structure
$\mathcal{M}$ is said to be {\em weakly $n$-spherically minimal}
if any definable (with parameters) subset of $M$ is a finite union
of convex sets.

A complete theory $T$ is said to be ({\em weakly}) {\em
$n$-spherically minimal} if all its models are (weakly)
$n$-spherically minimal.

Following \cite{KulMac} for $n=3$ we say on circular minimality
instead of $3$-spherical one.

\medskip
The following proposition immediately follows by Fact \ref{fact1}.

\begin{proposition}\label{prop1}
Let $\mathcal{M}=\langle M,K_n,\ldots\rangle$ be a {\rm
(}weakly{\rm )} $n$-spherically minimal structure, and
$a_1,\ldots,a_{n-2}$ be arbitrary pairwise distinct elements of
$M$. Take the relation
$\leq=\leq_{a_1,\ldots,a_{n-2}}=K_{n-2,a_1,\ldots,a_{n-2}}$ on
$$M_{a_1,\ldots,a_{n-2}}=M\setminus\{a_1,\ldots,a_{n-2}\}$$ and
$\leq'=\leq'_{a_1,\ldots,a_{n-2}}$ on $M$ as in Fact {\rm
\ref{fact1}}.

{\rm (i)} Let $\mathcal{M}_{a_1,\ldots,a_{n-2}}$ be the structure
with domain $M_{a_1,\ldots,a_{n-2}}$, the order $\leq$, and a
relation symbol for each $\{a_1,\ldots,a_{n-2}\}$-definable
relation of $\mathcal{M}$ on powers of $M_{a_1,\ldots,a_{n-2}}$.
Then $\mathcal{M}_{a_1,\ldots,a_{n-2}}$ is a {\rm (}weakly{\rm )}
$o$-minimal structure.

{\rm (ii)} Let $\mathcal{M}'_{a_1,\ldots,a_{n-2}}$ be the
structure with domain $M$, the order $\leq'$, and a relation
symbol for each $\{a_1,\ldots,a_{n-2}\}$-definable relation of
$\mathcal{M}$ on powers of $M$. Then
$\mathcal{M}'_{a_1,\ldots,a_{n-2}}$ is {\rm (}weakly{\rm )}
$o$-minimal.

\end{proposition}

Since $o$-minimal structures have $o$-minimal theories,
Proposition \ref{prop1} implies:

\begin{corollary}\label{cor1}
Any $n$-spherically minimal structure has $n$-spherically minimal
theory.
\end{corollary}

By the definition of $s_n(\mathcal{M})$ we have:

\begin{proposition}\label{prop2}
Let $\mathcal{M}=\langle M,\leq,\ldots\rangle$ be a weakly
$o$-minimal structure. Then $s_n(\mathcal{M})$ is a weakly
$n$-spherically minimal structure.
\end{proposition}

The following examples illustrate possibilities of spherical
minimality.

\begin{example} {\rm (cf. \cite[Example 2.9]{KulMac})}
{\rm Taking an $n$-dimensional sphere $S\subseteq\mathbb R^n$ of
radius $1$ and a $n$-spherical order $K_n$ on $S$ preserving
orientations of directed $n$-tetrahedrons we obtain the rotation
group ${\rm SO}(n)$ which preserves $K_n$. The expanded group
$\mathcal{G}=\langle {\rm SO}(n),K_n\rangle$ is {\em
$n$-spherically ordered}. Since $K_n$ is dense it is
$\omega$-categorical, with quantifier elimination in view of
\cite[Section 4]{KS4}. Similarly \cite[Theorem 5.1]{MacSt} the
structure $\mathcal{G}$ is $n$-spherically minimal.}
\end{example}

\begin{example} {\rm (cf. \cite[Example 2.10]{KulMac})}
{\rm Let $\mathcal{R}$ be an $o$-minimal expansion of a real
closed field $\langle R,\leq, +,\cdot\rangle$. For an ultrapower
${}^\ast\mathcal{R}$ of $\mathcal{R}$ and a set
$V:=\{x\in{}^\ast\mathcal{R}\mid |x|<n$ for some $n\in\omega\}$
the structure $\langle{}^\ast\mathcal{R},V\rangle$ is weakly
$o$-minimal by \cite{Baizh01}. Replacing the relation $<$ by a
correspondent $n$-spherical order $K_n$ on ${}^\ast\mathcal{R}$
and using Proposition \ref{prop2}, we obtain a weakly
$n$-spherically minimal structure
$\langle{}^\ast\mathcal{R},V,K_n\rangle$.}
\end{example}

\begin{example}
{\rm It is known \cite{PiSt, Edm} that $o$-minimal structures
$\langle M,\leq\rangle$ are elementary equivalent to ordered sets
of form $C_1+\ldots +C_m$, where $C_i$ are elementary equivalent
to one of the following ordered sets: a finite ordered set,
$\omega$, $\omega^\ast$, $\omega+\omega^\ast$,
$\omega^\ast+\omega$, $\mathbb Q$, and if $C_i$ does not have a
last element, then $C_{i+1}$ has a first element.}
\end{example}

Generalizing constructions in \cite{KulMac} and using Proposition
\ref{prop1} we spread possibilities for $o$-minimal structures
$\mathcal{M}$ both for circularly minimal structures
$c(\mathcal{M})=s_3(\mathcal{M})$ and for $n$-spherically minimal
structures $s_n(\mathcal{M})$, $n\geq 4$.

Example 2.12 in \cite{KulMac} illustrates that for the circularly
minimal structure $c(\omega+\omega^\ast+\mathbb
Q+\omega+\omega^\ast+\mathbb Q)$, ${\rm acl}(\emptyset)$ is a
proper superset of ${\rm dcl}(\emptyset)$ contrasting with the
$o$-minimal case, where ${\rm acl}(A)={\rm dcl}(A)$ for any subset
$A$ of a universe.

Similarly, the $n$-spherical structures, for $n\geq 4$, produce
algebraic closures ${\rm acl}(\emptyset)$ with ${\rm
acl}(\emptyset)\supsetneqq{\rm dcl}(\emptyset)$. These structures
are based on possibilities of distributions of discrete suborders,
of dense $n$-spherical orders described in \cite{KS4} with respect
to finitely many accumulation points for ``increasing/decreasing''
sequences of elements generalizing the order $\omega+\omega^\ast$.

\section{Topologies for spherically ordered theories}

\medskip
{\bf Definition.} For a $n$-spherically ordered structure
$\mathcal{M}=\langle M,K_n,\ldots\rangle$, $n\geq 4$, and for
elements $a_1,\ldots,a_n\in M$ we say that $a_n$ lays {\em
between} $a_1,\ldots,a_{n-1}$, or {\em inside} with respect to
$a_1,\ldots,a_{n-1}$ if
\begin{equation}\label{seq1}
\mathcal{M}\models K^0_n(a_1,\ldots,a_{n-1},a_n).
\end{equation}
If $n=2$ then the relation (\ref{seq1}) denotes that $a_2$ is {\em
greater} than $a_1$, and for $n=3$ that relation denotes that
$a_2$ lays {\em between} $a_1$ and $a_3$, or {\em inside} with
respect to $a_1,a_3$.

Similarly we say that $a_i$ lays {\em between} or {\em inside}
with respect to $a_1,\ldots,a_{i-1},a_{i+1},$ $\ldots,a_{n}$ if
$a_i$ lays {\em between} or {\em inside} with respect to the
circular permutation $a_{i+1},\ldots,$ $a_{n},a_1,\ldots,a_{i-1}$
of $a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_{n}$.

If $a_i$ does not lay between
$a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_{n}$ we say that $a_i$ lays
{\em outside} with respect to
$a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_{n}$.

By the axiom (nso1) of circular permutations the general case of
(non-)between\-ness is reduced to that relation with respect to
$a_1,\ldots,a_{n-1}$.

For a sequence $a_1,\ldots,a_{n-1}$ of pairwise distinct elements
the set ${\rm Int}(a_1,\ldots,a_{n-1})$ (respectively, ${\rm
Ext}(a_1,\ldots,a_{n-1})$) of all elements inside (outside) with
respect to $a_1,\ldots,a_{n-1}$ is called the {\em interior {\rm
(}exterior{\rm )}} with respect to $a_1,\ldots,a_{n-1}$.

The sets ${\rm Int}(a_1,\ldots,a_{n-1})$ and ${\rm
Ext}(a_1,\ldots,a_{n-1})$ are considered as neighbour\-hoods
$U_{\rm int}$ and $U_{\rm ext}$ of elements belonging to them.

By the definition we have $U_{\rm int}\cap U_{\rm ext}=\emptyset$
for the neighbourhoods $U_{\rm int}$ and $U_{\rm ext}$ with
respect to a fixed sequence $a_1,\ldots,a_{n-1}$.

Assuming $|M|\geq n$ each element $a\in M$ belongs to some
neighbourhoods $U_{\rm int}$ and $U_{\rm ext}$ with respect to
elements in $M\setminus\{a\}$. Thus the family $\mathcal{O}$
generated by the neighbourhoods with respect to unions and finite
intersections produces a topological space $\mathcal{X}_{\rm
Ext}^{\rm Int}(\mathcal{M})=(M,\mathcal{O})$. It is called {\em
$n$-spherical}.

The $n$-spherical topological space $\mathcal{X}_{\rm Ext}^{\rm
Int}(\mathcal{M})$ has two natural restrictions $\mathcal{X}^{\rm
Int}(\mathcal{M})$ and $\mathcal{X}_{\rm Ext}(\mathcal{M})$
generated by neighbourhoods $U_{\rm int}$ and $U_{\rm ext}$,
respectively. These re\-stric\-tions are called {\em
$n$-spherical}, too.

\begin{note}\label{dual}
{\rm Since neighbourhoods $U_{\rm int}$ and $U_{\rm ext}$ with
respect to a fixed sequence $a_1,\ldots,a_{n-1}$ are disjoint and
$U_{\rm int}\cup U_{\rm ext}=M\setminus\{a_1,\ldots,a_{n-1}\}$
these neighbourhoods are complement each other. Thus
$\mathcal{X}^{\rm Int}(\mathcal{M})$ and $\mathcal{X}_{\rm
Ext}(\mathcal{M})$ are dual: one of them is uniquely defined by
another one, via complements. Moreover, both $\mathcal{X}^{\rm
Int}(\mathcal{M})$ and $\mathcal{X}_{\rm Ext}(\mathcal{M})$,
separately and together, define $\mathcal{X}_{\rm Ext}^{\rm
Int}(\mathcal{M})$.}
\end{note}

\begin{proposition}\label{PropHaus}
For any $n$-spherically ordered structure $\mathcal{M}$ with
$|M|\geq n$ the space $\mathcal{X}_{\rm Ext}^{\rm
Int}(\mathcal{M})$ is Hausdorff.
\end{proposition}

Proof. Let $a$ and $b$ be two distinct elements in $M$.

If $n=2$ then we have $a<b$ or $b<a$. Without loss of generality
we assume that $a<b$. Having $c\in M$ with $a<c<b$ we obtain
disjoint neighbourhoods ${\rm Int}(c)$ and ${\rm Ext}(c)$
separating $a\in {\rm Ext}(c)$ and $b\in {\rm Int}(c)$. If $c$
does not exist, i.e., $b$ is the successor of $a$ we take
separating neighbourhoods ${\rm Int}(a)$ and ${\rm Ext}(b)$ with
$a\in {\rm Ext}(b)$ and $b\in {\rm Int}(a)$. Thus
$\mathcal{X}_{\rm Ext}^{\rm Int}(\mathcal{M})$ is Hausdorff.

Now let $n\geq 3$. We fix pairwise distinct elements
$d_1,\ldots,d_{n-2}\in M\setminus\{a,b\}$. We have two
possibilities for $a$ and $b$ with respect to
$d_1,\ldots,d_{n-2}$: $$\models K^0_n(d_1,\ldots,d_{n-2},a,b),$$
or $\models \neg K^0_n(d_1,\ldots,d_{n-2},a,b)$, i.e., $\models
K^0_n(d_1,\ldots,d_{n-2},b,a)$. Following Fact \ref{fact1} the
relation $K^0_n(d_1,\ldots,d_{n-2},x,y)$ defines a linear order
$<$ on $M\setminus\{d_1,\ldots,d_{n-2}\}$ imply\-ing $a<b$ or
$b<a$. Now repeating the arguments for $n=2$ we obtain a pair of
disjoint neighbourhoods ${\rm Int}(d_1,\ldots,d_{n-2},c)$ and
${\rm Ext}(d_1,\ldots,d_{n-2},c)$, or ${\rm Int}(d_1,\ldots,$
$d_{n-2},a)$ and ${\rm Ext}(d_1,\ldots,d_{n-2},b)$, or ${\rm
Ext}(d_1,\ldots,d_{n-2},a)$ and ${\rm Int}(d_1,\ldots,d_{n-2},b)$,
separating $a$ and $b$ and witnessing the Hausdorffness of
$\mathcal{X}_{\rm Ext}^{\rm Int}(\mathcal{M})$.
\endproof

\begin{note}\label{nlo} {\rm Any $2$-spherically ordered structure
$\mathcal{M}$ with $|M|\geq 2$, i.e., a linearly ordered structure
with an order $K_2=\,\,\leq$ of cardinality at least $3$, produces
the $T_0$-spaces $\mathcal{X}^{\rm Int}(\mathcal{M})$ and
$\mathcal{X}_{\rm Ext}(\mathcal{M})$ which are not $T_1$-spaces.
Indeed, there are $a_1,a_2\in M$ with $a_1<a_2$, and for any $a\in
M$ we have ${\rm Int}(a)=\{b\in M\mid a<b\}$, ${\rm Ext}(a)=\{b\in
M\mid b<a\}$. Thus, the neighbourhoods $U_{\rm int}$, respectively
$U_{\rm ext}$, form chains isomorphic to $<$ and witnessing that
$\mathcal{X}^{\rm Int}(\mathcal{M})$ and $\mathcal{X}_{\rm
Ext}(\mathcal{M})$ are $T_0$-spaces, not $T_1$-ones. In
particular, $a_2$ is separated from $a_1$ by ${\rm Int}(a_1)$ but
$a_1$ is not separated from $a_2$ by interiors. Similarly, $a_1$
is separated from $a_2$ by ${\rm Ext}(a_2)$ but $a_2$ is not
separated from $a_1$ by exteriors.}
\end{note}

\begin{note}\label{nco} {\rm Any $3$-spherically ordered structure
$\mathcal{M}$ with $|M|\geq 3$, i.e., a circularly ordered
structure with an order $K_3$, produces the $T_2$-spaces
$\mathcal{X}^{\rm Int}(\mathcal{M})$ and $\mathcal{X}_{\rm
Ext}(\mathcal{M})$. Indeed, taking distinct elements $a_1,a_2\in
M$ we have either $\models K^0_3(a_1,a_2,b)$ or $\models
K^0_3(a_2,a_1,b)$ for some $b$.

In the first case we have $a_2\in{\rm Int}(a_1,b)$. If there is an
element $c$ between $a_1$ and $a_2$ we choose an element $d$
between $b$ and $a_1$, or taking $d=b$ if there are no elements
between $b$ and $a_1$, producing distinct neighbourhoods ${\rm
Int}(d,c)$ and ${\rm Int}(c,b)$ containing $a_1$ and $a_2$,
respectively, and witnessing that $\mathcal{X}^{\rm
Int}(\mathcal{M})$ is Hausdorff. If there are no elements between
$a_1$ and $a_2$ we repeat the choice of $d$ and obtain distinct
neighbourhoods ${\rm Int}(d,a_2)$ and ${\rm Int}(a_1,b)$
containing $a_1$ and $a_2$, respectively, witnessing again the
Hausdorffness of $\mathcal{X}^{\rm Int}(\mathcal{M})$.

In the second case we have similarly $a_1\in{\rm Int}(a_2,b)$ and
there are $c,d\in M$ with distinct neighbourhoods ${\rm Int}(d,c)$
and ${\rm Int}(c,b)$ containing $a_2$ and $a_1$, respectively, or
distinct neighbourhoods ${\rm Int}(d,a_1)$ and ${\rm Int}(a_2,b)$
containing $a_2$ and $a_1$, respectively, and witnessing again the
Hausdorffness of $\mathcal{X}^{\rm Int}(\mathcal{M})$.

The Hausdorffness of $\mathcal{X}_{\rm Ext}(\mathcal{M})$ is
implied by its duality with respect to the Hausdorff space
$\mathcal{X}^{\rm Int}(\mathcal{M})$.}
\end{note}

\begin{note}\label{nso} {\rm Any $n$-spherically ordered structure
$\mathcal{M}$ with $|M|\geq n\geq 4$ produces the $T_2$-spaces
$\mathcal{X}^{\rm Int}(\mathcal{M})$ and $\mathcal{X}_{\rm
Ext}(\mathcal{M})$. Indeed, taking distinct elements $a_1,a_2\in
M$ we choose pairwise distinct elements $b_1,\ldots,b_{n-3}\in
M\setminus\{a_1,a_2\}$ and obtain a circular order
$K^0_n(b_1,\ldots,b_{n-3},x,y,z)$ on $M\setminus\{a_1,a_2\}$
implying the Hausdorffness for $\mathcal{X}^{\rm
Int}(\mathcal{M})$ and $\mathcal{X}_{\rm Ext}(\mathcal{M})$ in
view of Remark \ref{nco}.}
\end{note}

\begin{note}\label{triv} {\rm Any $n$-spherically ordered structure
$\mathcal{M}$ with $|M|<n$ produces the spaces $\mathcal{X}^{\rm
Int}(\mathcal{M})$, $\mathcal{X}_{\rm Ext}(\mathcal{M})$,
$\mathcal{X}_{\rm Ext}^{\rm Int}(\mathcal{M})$ which are not
$T_0$-spaces. Indeed, if $|M|<n$ then $K_n=M^n$ implying that
there are no separated elements at all.}
\end{note}

Summarizing the assertions and remarks above we obtain the
following theorems.

\begin{theorem}\label{th01}
For any $n$-spherically ordered structure $\mathcal{M}$ the
following conditions are equivalent:

$(1)$ $\mathcal{X}^{\rm Int}(\mathcal{M})$ is a $T_0$-space;

$(2)$ $\mathcal{X}_{\rm Ext}(\mathcal{M})$ is a $T_0$-space;

$(3)$ $\mathcal{X}_{\rm Ext}^{\rm Int}(\mathcal{M})$ is Hausdorff;

$(4)$ $|M|\geq n$.
\end{theorem}

Proof. $(1)\Rightarrow(4)$, $(2)\Rightarrow(4)$,
$(3)\Rightarrow(4)$ follow by Remark \ref{triv}.
$(4)\Rightarrow(1)$ and $(4)\Rightarrow(2)$ are implied by Remark
\ref{nlo}, $(4)\Rightarrow(3)$ holds by Proposition
\ref{PropHaus}. \endproof

\begin{theorem}\label{th02}
For any $n$-spherically ordered structure $\mathcal{M}$ the
following conditions are equivalent:

$(1)$ $\mathcal{X}^{\rm Int}(\mathcal{M})$ is a $T_1$-space;

$(2)$ $\mathcal{X}_{\rm Ext}(\mathcal{M})$ is a $T_1$-space;

$(3)$ $\mathcal{X}^{\rm Int}(\mathcal{M})$ is Hausdorff;

$(4)$ $\mathcal{X}_{\rm Ext}(\mathcal{M})$ is Hausdorff;

$(5)$ $|M|\geq n\geq 3$.
\end{theorem}

Proof. $(1)\Rightarrow(5)$, $(2)\Rightarrow(5)$,
$(3)\Rightarrow(5)$, $(4)\Rightarrow(5)$ follow by Remark
\ref{triv}. $(5)\Rightarrow(3)$ and $(5)\Rightarrow(4)$ are
implied by Remarks \ref{nco} and \ref{nso}, and
$(3)\Rightarrow(1)$, $(4)\Rightarrow(2)$, $(1)\Leftrightarrow(2)$
hold by the definition.
\endproof

\section{Definable minimality for linearly ordered and spherically ordered theories}

{\bf Definition} \cite{BMN, Ta11}. A structure $\mathcal{M}$ is
called {\em definably minimal} if each definable, with parameters,
subset of $M$ is either finite or cofinite.

\medskip
{\bf Definition} \cite{BaLa}.  A structure $\mathcal{M}$ is called
{\em strongly minimal} if each structure
$\mathcal{N}\equiv\mathcal{M}$ is definably minimal.

\medskip
We denote by ${\rm Mod}(T)$ the class of all $T$-models and by
${\rm Min}(T)$ the class of all definably minimal $T$-models.

By the definition we have the following:

\begin{proposition}
A theory $T$ is strongly minimal if and only if ${\rm Mod}(T)={\rm
Min}(T)$.
\end{proposition}

The linear order $\omega$ represents a definably minimal structure
which is not strongly minimal, since the theory ${\rm Th}(\omega)$
is unstable, with the strict order property.

Following \cite{BMN, Ta11} definably minimal linear orders are
exhausted by expansions of $m$, $\omega$, $\omega^\ast$,
$\omega+\omega^\ast$, $\omega+m$, $m+\omega^\ast$ for natural $m$.

Here the only orders $m$ are minimal models of their theories,
under inclusion, since $\omega\simeq 1+\omega$ and
$\omega^\ast\simeq \omega^\ast+1$. Thus unstable definably minimal
linear orders do not have minimal models under inclusion. These
minimal models and their hypergraphs are studied both in general
context \cite{Su08} and for theories of abelian groups
\cite{Deiss, Su20}.

Using the description of definably minimal linear orders we
observe that there are no uncountable definably minimal linear
orders whereas by L\"{o}wenheim-Skolem theorem strongly minimal
theories with infinite models have definably minimal models of
arbitrarily large infinite cardinality.

The following theorem gives a description for definably minimal
extensions and restrictions of definably minimal linear orders.

\begin{theorem}\label{dmlo1}
For any linear definably minimal order $\mathcal{M}$ a linear
order $\mathcal{N}\supset\mathcal{M}$ with finite $N\setminus M$
is definably minimal if and only if $\mathcal{N}$ does not have
the form $\omega+m+\omega^\ast$ for some natural $m>0$.
\end{theorem}

Proof. Let $\mathcal{N}\supset\mathcal{M}$ be a linear order with
finite $Z=N\setminus M$. We have the following possibilities:

i) $\mathcal{M}=m\in\omega$ and elements of $Z$ are situated
before, between and/or after elements of $m$ producing the finite
definably minimal linear order with $m+|Z|$ elements;

ii) $\mathcal{M}=\omega$ and elements of $Z$ are situated before,
between and/or after elements of $\omega$ obtaining the definably
minimal order $\omega+m$, where $m>0$ iff there are elements of
$Z$ after all elements of $\omega$;

iii) $\mathcal{M}=\omega^\ast$ and elements of $Z$ are situated
before, between and/or after elements of $\omega^\ast$ obtaining
the definably minimal order $m+\omega^\ast$, where $m>0$ iff there
are elements of $Z$ before all elements of $\omega^\ast$;

iv) $\mathcal{M}=\omega+\omega^\ast$ and elements of $Z$ are
situated before, between and/or after elements of
$\omega+\omega^\ast$ producing either isomorphic definably minimal
order $\omega+\omega^\ast$, if $Z$ does not contain elements
greater than all elements of $\omega$ and less than all elements
of $\omega^\ast$, or an order $\omega+m+\omega^\ast$, if $Z$
contains an element $a$ greater than all elements of $\omega$ and
less than all elements of $\omega^\ast$; this element $a$ divides
$M$ into two infinite parts by the formulae $x<a$ and $a\leq x$
violating the definable minimality;

v) $\mathcal{M}=\omega+m$ and elements of $Z$ are situated before,
between and/or after elements of $\omega+m$ obtaining the
definably minimal order $\omega+m'$, where $m'\geq m$ and $m'>m$
iff there are elements of $Z$ after all elements of $\omega+m$;

vi) $\mathcal{M}=m+\omega^\ast$ and elements of $Z$ are situated
before, between and/or after elements of $m+\omega^\ast$ obtaining
the definably minimal order $m'+\omega^\ast$, where $m'\geq m$ and
$m'>m$ iff there are elements of $Z$ before all elements of
$m+\omega^\ast$.

The only case iv) with $\omega+m+\omega^\ast$ fails the definable
minimality for $\mathcal{N}$. \endproof

\begin{theorem}\label{dmlo2} For any linear definably minimal order $\mathcal{M}$ a linear
order $\mathcal{N}\supset\mathcal{M}$ with countable $N\setminus
M$ is definably minimal if and only if $\mathcal{N}$ does not
contain forms $\omega^\ast+\omega$, $\omega^\ast+\omega^\ast$,
$\omega+\omega$, $\omega+m+\omega^\ast$ for some natural $m>0$.
\end{theorem}

Proof. Clearly each type $\omega^\ast+\omega$,
$\omega^\ast+\omega^\ast$, $\omega+\omega$,
$\omega+m+\omega^\ast$, for some natural $m>0$, fails the
definable minimality. Indeed, we have the formulae $x<a$ and
$a\leq x$ with infinitely many solutions, where $a\in\omega$ for
the type $\omega^\ast+\omega$, $a$ belongs to the first
$\omega^\ast$ in $\omega^\ast+\omega^\ast$, $a$ belongs to the
second $\omega$ in $\omega+\omega$, $a\in m$ for
$\omega+m+\omega^\ast$. Conversely, if $\mathcal{N}$ is definably
minimal then taking an arbitrary element $a\in N$ we have either
finite $\mathcal{N}<a$ and countable $a<\mathcal{N}$, or countable
$\mathcal{N}<a$ and finite $a<\mathcal{N}$.

In the first case for any $b>a$ either there are finitely many
elements between $a$ and $b$ and countably many elements greater
than $b$, or there are countably many elements between $a$ and $b$
and finitely many elements greater than $b$. Therefore
$\mathcal{N}$ has the type $m+\omega^\ast$, if $a\in m$, or
$\omega$, for $a<b\in\omega$, or $\omega+m$, if $a\in\omega$, and
$b\in\omega$ or $b\in m$, or $\omega+\omega^\ast$, if
$a\in\omega$, and $b\in\omega$ or $b\in\omega^\ast$.

In the second case we replace $<$ by $>$ and obtain the following
possibilities:  $m+\omega^\ast$, $\omega^\ast$, $\omega+m$,
$\omega+\omega^\ast$.

In any case $\mathcal{N}$ does not contain forms
$\omega^\ast+\omega$, $\omega^\ast+\omega^\ast$, $\omega+\omega$,
$\omega+m+\omega^\ast$. \endproof

\begin{theorem}\label{dmlo3} For any linear definably minimal order $\mathcal{M}$ each
restriction $\mathcal{N}\subset\mathcal{M}$ is definably minimal,
too.
\end{theorem}

Proof. We have the following possibilities for $\mathcal{N}$:

i) $\mathcal{M}=m\in\omega$ and $\mathcal{N}\simeq m'$ for $m'\leq
m$;

ii) $\mathcal{M}=\omega$, and $\mathcal{N}\simeq\omega$, if $N$ is
infinite, or $\mathcal{N}\simeq m\in\omega$, if $N$ is finite;

iii) $\mathcal{M}=\omega^\ast$, and
$\mathcal{N}\simeq\omega^\ast$, if $N$ is infinite, or
$\mathcal{N}\simeq m\in\omega$, if $N$ is finite;

iv) $\mathcal{M}=\omega+\omega^\ast$, and
$\mathcal{N}\simeq\omega+\omega^\ast$, if $N$ has infinitely many
elements of $\omega$ and infinitely many elements of
$\omega^\ast$; $\mathcal{N}\simeq\omega+m$, if $N$ has infinitely
many elements of $\omega$ and finitely many elements of
$\omega^\ast$; $\mathcal{N}\simeq m+\omega^\ast$, if $N$ has
finitely many elements of $\omega$ and infinitely many elements of
$\omega^\ast$; $\mathcal{N}\simeq m$, if $N$ has finitely many
elements of $\omega$ and finitely many elements of $\omega^\ast$;

v) $\mathcal{M}=\omega+m$, and $\mathcal{N}\simeq\omega+m'$ for
$m'\leq m$, if $N$ is infinite, or $\mathcal{N}\simeq
m'\in\omega$, if $N$ is finite;

vi) $\mathcal{M}=m+\omega^\ast$, and $\mathcal{N}\simeq
m'+\omega^\ast$ for $m'\leq m$, if $N$ is infinite, or
$\mathcal{N}\simeq m'\in\omega$, if $N$ is finite.

Thus all possible restrictions $\mathcal{N}$ of $\mathcal{M}$ are
again definably minimal. \endproof

\medskip
Theorems \ref{dmlo1}, \ref{dmlo2}, \ref{dmlo3} are naturally
spread for theories of $n$-spherical orders, via the
$n$-spherifications.

\medskip
The following definition modifies the notion of definable
minimality for $n$-spherical theories.

\medskip
{\bf Definition.} A $n$-spherically ordered structure
$\mathcal{M}$ is called {\em $K_n$-minimal} if for any pairwise
distinct elements $a_1,\ldots,a_{n-1}\in M$ either ${\rm
Int}(a_1,\ldots,a_{n-1})$ or ${\rm Ext}(a_1,\ldots,$ $a_{n-1})$ is
finite, i.e., $M$ is divided by $K_n$ into a finite and a cofinite
parts with respect to any distinct $a_1,\ldots,a_{n-1}$.

We say that $\mathcal{M}$ produces a {\em limit point}, or an {\em
accumulation point} $a$, in a model of ${\rm Th}(\mathcal{M})$,
with respect to $K_n$ if there is a sequence
$(\overline{a}_m)_{m\in\omega}$, where each $\overline{a}_m$
consists of $n-1$ pairwise distinct coordinates, and a sequence
$(U_m)_{m\in\omega}$ of neighbourhoods of the form ${\rm
Int}(\overline{a}_m)$ or ${\rm Ext}(\overline{a}_m)$ such that the
sets $V_m=U_0\cap\ldots\cap U_m$ form a strictly decreasing chain
with ${\rm tp}(a/M)\ni\varphi_m(x,\overline{a}_m)$, where
$\varphi_m(x,\overline{a}_m)$ is a formula
$K^0_n(x,\overline{a}_m)$ or $\neg
K^0_n(x,\overline{a}_m)\wedge\bigwedge\limits_{a'\in\overline{a}_m}\neg
x\approx a'$ with the set $U_m$ of solutions, $m\in\omega$.

\medskip
\begin{note}\label{K2min} {\rm Following \cite{BMN, Ta11} $K_2$-minimal structures are
exhausted by expansions of $m$, $\omega$, $\omega^\ast$,
$\omega+\omega^\ast$, $\omega+m$, $m+\omega^\ast$ for natural $m$.
In every one of these cases accumulation points are equivalent:
they have the same type over the model, i.e., the quotient with
respect to the equivalence relation $x\equiv y$ is a singleton.
Here the accumulation points are produced by the neighbourhoods
$U_{\rm int}$ for $\omega$, $U_{\rm ext}$ for $\omega^\ast$,
$U_{\rm ext}\cap U_{\rm int}$ for $\omega+\omega^\ast$,
$\omega+m$, $m+\omega^\ast$. Thus each accumulation point $a$ is a
limit of an increasing or a decreasing sequence.

The same possibilities replacing the linear orders $m$, $\omega$,
$\omega^\ast$, $\omega+\omega^\ast$, $\omega+m$, $m+\omega^\ast$
by the circular ones $c(\cdot)$ are valid for $K_3$-minimal
structures. In fact we have three non-isomorphic possibilities
$c(\omega)$, $c(\omega^\ast)$, and $c(\omega+\omega^\ast)$ with
accumulation points, since $c(\omega+m)\simeq c(\omega)$ and
$c(m+\omega^\ast)\simeq c(\omega^\ast)$. Moreover, $c(\omega)$ and
$c(\omega^\ast)$ are mutually dual and connected by an
anti-isomorphism between $\omega$ and $\omega^\ast$, and
$c(\omega+\omega^\ast)$ is self-dual: it is anti-isomorphic to
itself.

$K_n$-minimal structures have similar possibilities combining
interiors and exteriors. Here for
$K^0_n(a_1,\ldots,a_{i-1},x,a_{i+1},\ldots,a_n)$ both some
elements $a_j$ may be fixed and some $a_k$ may be moving producing
cases above, if $n-2$ elements are fixed, and additional cases
with at least two moving points.

For instance, taking $K^0_4(a^0_1,a^0_2,a^0_3,x)$ we can consider
a sequence $$(\langle a^m_1,a^m_2,a^m_3\rangle)_{m\in\omega}$$
with $a^{m+1}_1,a^{m+1}_2,a^{m+1}_3\in{\rm
Int}(a^m_1,a^m_2,a^m_3)$ (respectively, with
$a^{m+1}_1,a^{m+1}_2,a^{m+1}_3\in{\rm Ext}(a^m_1,a^m_2,a^m_3)$)
and an accumulation point $a\in{\rm Int}(a^m_1,a^m_2,a^m_3)$
($a'\in{\rm Int}(a^m_1,a^m_2,$ $a^m_3)$), $m\in\omega$,
illustrating the $K_n$-minimalities.}
\end{note}

Applying the correspondence between linear and spherical orders
described in Fact \ref{fact1} and Hausdorffness of topology with
respect to $\mathcal{X}_{\rm Ext}^{\rm Int}(\mathcal{M})$ in
Theorem \ref{th01} we obtain the following possibilities for
$K_n$-minimal structures with $n\geq 2$.

\begin{theorem}\label{thsphmin} For any infinite $n$-spherically ordered structure
$\mathcal{M}=\langle M,K_n,\ldots\rangle$ the following conditions
are equivalent:

{\rm (1)} $\mathcal{M}$ is $K_n$-minimal;

{\rm (2)} $\mathcal{M}$ produces unique accumulation point with
respect to $K_n$;

{\rm (3)} any linearization $\leq$ of $\mathcal{M}$ is
$\leq$-minimal;

{\rm (4)} any linearization $\leq$ of $\mathcal{M}$ produces
unique limit point with respect to $\leq$.
\end{theorem}

Proof. $(1)\Rightarrow(2)$. Since $\mathcal{M}$ is infinite there
are infinitely many neighbourhoods producing at least one
accumulation point. If $\mathcal{M}$ produces at least two
accumulation points $a$ and $b$ with respect to $K_n$. Since the
topology $\mathcal{X}_{\rm Ext}^{\rm Int}(\mathcal{M})$ is
Hausdorff by Theorem \ref{th01}, there are disjoint infinite
neighbourhoods $U_1$ and $U_2$ separating $a$ and $b$. Both the
formula $K_n(\ldots)$ for $U_1$ and its negation $\neg
K_n(\ldots)$ have infinitely many solutions contradicting the
$K_n$-minimality of $\mathcal{M}$.

$(2)\Rightarrow(1)$. If $\mathcal{M}$ is not $K_n$-minimal then
there are infinite neighbourhoods $U_1$ and $U_2$ defined by a
formula $K_n(\ldots)$ and its negation $\neg K_n(\ldots)$,
respectively. Since the topology $\mathcal{X}_{\rm Ext}^{\rm
Int}(\mathcal{M})$ is Hausdorff both the neighbourhood $U_1$ and
the neighbourhood  $U_2$ produce accumulation points $a$ and $b$.
These points are distinct since $U_1$ and $U_2$ are disjoint.

$(3)\Leftrightarrow(4)$ follows by $(1)\Leftrightarrow(2)$ for
$n=2$.

$(1)\Rightarrow(3)$. Let $\mathcal{M}$ be $K_n$-minimal, $\leq$ be
a linearization of $\mathcal{M}$ with respect to
$$K_n(a_1,\ldots,a_{n-2},x,y).$$ Taking an arbitrary element $b\in
M\setminus\{a_1,\ldots,a_{n-2}\}$, by the $K_n$-minimality of
$\mathcal{M}$, we have finitely or cofinitely many solutions for
$K_n(a_1,\ldots,a_{n-2},b,y)$ and for
$K_n(a_1,\ldots,a_{n-2},x,b)$, i.e.,  for $b\leq y$ and for $x\leq
b$ confirming that the linearization is $\leq$-minimal.

$(3)\Rightarrow(1)$. Let any linearization $\leq$ of $\mathcal{M}$
be $\leq$-minimal. Taking a formula
$$K_n(a_1,\ldots,a_{n-2},a_{n-1},y)$$ and a $\leq$-minimal
linearization $\leq$ defined by the formula
$K_n(a_1,\ldots,a_{n-2},x,y)$ we have finitely or cofinitely many
solutions for $a_{n-1}\leq y$ and for $x\leq a_{n-1}$ implying
that $K_n(a_1,\ldots,a_{n-2},a_{n-1},y)$ has finitely or
cofinitely many solutions and confirming that $\mathcal{M}$ is
$K_n$-minimal. \endproof

\medskip
\begin{note}{\rm
It is essential in Theorem \ref{thsphmin} that {\em all}
linearizations $\leq$ of $\mathcal{M}$ are $\leq$-minimal. Indeed,
the linear order $\omega+1+\omega^\ast$ is not definably minimal
producing $n$-spherical orders $K_n$ which are not $K_n$-minimal,
whereas the linearization $\leq'$ removing the element $1$ becomes
$\leq'$-minimal.

We observe a similar effect taking a $n$-spherical order $K_n$
with $m<n$ accumulation points and transforming it into a
$\leq$-minimal linearization by removing $n-2$ elements, say with
$\widehat{K}_n(\omega+(n-2)+\omega^\ast)$.}
\end{note}

\section{Convexity rank for spherically ordered theories}

The following notion of convexity rank generalizes that known one
from the class of linearly ordered theories \cite{KS_JSL} to the
classes of $n$-spherically ordered theories. It produces a measure
of minimality with respect to convex sets.

\medskip
{\bf Definition.}  Let $T$ be a weakly $n$-spherically mi\-ni\-mal
theory, $\mathcal{M}$ be a sufficiently saturated model of $T,$
$\varphi(x)$ be a $M$-definable formula with one free variable.
The {\it convexity rank of the formula} $\varphi(x)$ $({\rm
RC}(\varphi(x)))$ is defined as follows:

0) ${\rm RC}(\varphi(x))=-1$ if $\varphi(M)$ is empty, and ${\rm
RC}(\varphi(x)) \geq 0$ if $|\varphi(M)|\in\omega\setminus\{0\}$;

1)  ${\rm RC}(\varphi(x)) \geq 1$ if $\varphi(M)$ is infinite;

2) ${\rm RC}(\varphi(x)) \geq  \alpha  +  1$ if there is a
parametrically definable equivalence relation $E(x,y)$ and
infinitely many elements $b_i, i\in \omega $, such that:
\begin{itemize}
\item for any distinct $i, j\in \omega$,
$\mathcal{M} \models \neg E(b_i, b_j)$;
\item for any $i\in \omega$, ${\rm RC}(E(x,b_i))\geq \alpha$ and $E(\mathcal{M},b_i)$ is a convex subset of
$\varphi(\mathcal{M})$;
\end{itemize}

3) ${\rm RC}(\varphi(x)) \geq \delta$ if ${\rm RC}(\varphi(x))
\geq \alpha$ for all $\alpha< \delta \; (\delta$ is a limit
ordinal).

\medskip
If ${\rm RC}(\varphi(x)) = \alpha$ for some $\alpha$ then we say
that ${\rm RC}(\varphi(x))$ is {\em defined}. Otherwise, i.e., if
${\rm RC}(\varphi(x))\geq\alpha$ for all $\alpha$, we put ${\rm
RC}(\varphi(x)) =\infty$.

\medskip
The {\it convexity rank of $1$-type} $p$ (${\rm RC}(p)$) is the
infimum of the set $\{{\rm RC}(\varphi(x)) \mid \varphi(x)\in
p\}$, i.e., ${\rm RC}(p):= {\rm inf} \{{\rm RC}(\varphi(x)) \mid
\varphi(x)\in p\}$.

For the theory $T$ the {\em convexity rank} ${\rm RC}(T)$ is the
supremum of values ${\rm RC}(p)$ of convexity ranks of $1$-types
$p$.

\begin{note}\label{rem_lin_CR}
{\rm The operator $\widehat{L}$ of linearization with respect to
elements $a_1,\ldots,a_{n-2}$ preserves the convexity rank ${\rm
RC}(T)$ of a theory $T$, if ${\rm RC}(T)$ is defined without these
elements. Conversely, under that condition the operator
$\widehat{K}_n$ of $n$-spherification preserves the convexity
rank, too.

At the same time, extensions of ordered structures by new elements
can produce additional definable relations increasing the
convexity rank.}
\end{note}

\begin{note}\label{rem_lin_CR1}
{\rm Similarly the connection between $o$-minimality and weak
$o$-mi\-ni\-ma\-li\-ty, $n$-spherically minimality of a theory $T$
means that $T$ is weakly $n$-spherically minimal with ${\rm
RC}(T)=1$.}
\end{note}

\begin{theorem}\label{th_cr_spher}
For any natural $m\geq 1$ and $n\geq 2$ there is a countably
categorical weakly $n$-spherically minimal theory $T_{m,n}$ such
that ${\rm RC}(T_{m,n})=m$.
\end{theorem}

Proof. There are two ways to construct a required theory
$T_{m,n}$. The first one is based on examples of countably
categorical weakly $o$-minimal theories $T_{m,2}$, \cite[Section
2]{HMMNT}, \cite[Example 1.2]{KS9}, being expansions of dense
linear orders by refinements of definable equivalence classes
using new binary symbols. The theory $T_{m,2}$ is transformed into
$T_{m,n}$ by $n$-spherification $\widehat{K}_n$ preserving the
convexity rank in view of Remark \ref{rem_lin_CR}.

Another way is based directly on dense $n$-spherical order $K_n$
\cite{KS4}. We introduce $m$ new symbols $E_1,\ldots,E_m$ for
equivalence relations, with $E_1\supset E_2\supset\ldots\supset
E_m$ having convex equivalence classes only. We divide the
universe into countably many $E_1$-classes, each $E_1$-class is
divided into countably many $E_2$-classes, etc., continuing the
process $m$ times and preserving the $\omega$-categoricity for the
resulting theory $T_{m,n}$. \endproof

\section{Conclusion}

We studied natural modifications of $o$-minimality, weak
$o$-minimality, definable minimality adapted for spherical
theories and producing $n$-spherical minimality, weak
$n$-spherical minimality, definable minimality for the class of
$n$-spherical struc\-tu\-res and their theories. Topological and
definable properties for various spherical minimalities are
described. The values of convexity rank for countably categorical
weakly $n$-spherically minimal theories are found. Thus the
possibilities of minimality conditions, of topologies and of ranks
are described for the spherical case.

We show that there are links between linear and spherical orders
via the operators of linearization and of spherification. At the
same time spherical orders produce new properties with respect to
linear ones. In particular, spherical orders admit non-trivial
algebraic closures which differ from definable one whereas these
closures coincide for linearly ordered structures.

It would be natural to transform the well developed, rich and
productive clas\-si\-fi\-ca\-ti\-on theory of $o$-minimality and
its variation for linearly and partially ordered structures to the
class of spherically ordered theories using possibilities of
minimality conditions and the convexity rank. Thus the problem
arises on a transformation of the descriptions of linearly and
circularly ordered structures with minimality, topology and rank
conditions till spherically ordered ones.

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\end{document}
