On One Laura Mayer’s Theorem

Abstract

The article’s primary focus is on the study of the number of countable non-isomorphic models of linearly ordered theories. The orthogonality of 1-types and their convex closures is employed to analyse a class of theories with a specific type of monotonic non-orthogonality, which includes weakly o-minimal theories. For such theories, a theorem analogous to L. Mayer’s result on the independence of any pairwise independent family of 1-types in o-minimal theories is proven. The article provides conditions for the infinity and maximality of the countable spectrum of weakly o-minimal theories.

1. Introduction

Definition 1

([1]). A linearly ordered structure $\mathfrak{M}=\langle M;<,\dots \rangle$ is o-minimal if all of its subsets definable with parameters are finite unions of points and open intervals. A theory is o-minimal if all its models are o-minimal.

When proving Vaught’s conjecture for o-minimal theories, L. Mayer confirmed the following theorem.

Theorem 1

([2]). Let T be an o-minimal theory such that T has fewer than $2^{{\aleph }_0}$ countable models. Let Γ be a pairwise independent set of nonisolated elements of $S_1(\varnothing )$. Then Γ is independent.

L. Mayer’s work is based on the theory of non-weak orthogonality which was developed for o-minimal theories by D. Marker [3]. In this article, we use ideas from the proof of L. Mayer’s theorem to give its generalization for small theories with monotonic non-orthogonality (Theorem 4). While in o-minimal theories the proof relies on independence of 1-types, we use orthogonality of convex closures of 1-types.

According to S. Shelah, two complete types are weakly orthogonal, if their union is a complete type [4]. The following is an equivalent definition of weak orthogonality of complete 1-types.

Let $A\subseteq M$, $p,q\in S_1\left(A\right)$, and $\mathfrak{M}$ be an ${\left|A\right|}^{+}$-saturated structure of a language $\mathcal{L}$.

Definition 2.

The type p is not weakly orthogonal to the type q, $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$, if there exist an A-definable formula $H(x,y)$, $\alpha \in p\left(\mathfrak{M}\right)$, and realizations ${\beta }_1,{\beta }_2\in q\left(\mathfrak{M}\right)$ such that ${\beta }_1\in H(\mathfrak{M},\alpha )$ and ${\beta }_2\notin H(\mathfrak{M},\alpha )$.

The rations of weak and almost orthogonality of 1-types are equivalence relations in weakly o-minimal theories, in particular, they are symmetric.

Definition 3

([5]). A linearly ordered structure is weakly o-minimal if all of its parametrically definable subsets are finite unions of convex sets. A theory is weakly o-minimal if all its models are weakly o-minimal.

Vaught’s conjecture was confirmed for the class of quite o-minimal theories [6], for the class of weakly o-minimal theories of convexity rank 1 [7], and later, in [8], for weakly o-minimal theories of finite convexity rank. During the proofs, analogues of Theorem 1 were proved for (weakly) orthogonal families of types in the three aforementioned classes of theories.

In o-minimal and weakly o-minimal theories, each complete 1-type has a convex set of all realizations. In [9], the concepts of convex weak and convex almost orthogonality of convex 1-types in linearly ordered theories were introduced as generalizations of weak and almost orthogonality of 1-types in weakly o-minimal theories. The relation of non-weak convex orthogonality is an equivalence relation in theories with monotonic non-orthogonality on equivalence classes (Corollary 2).

Theorem 4 implies that if in a theory with monotonic non-orthogonality on equivalence classes there exists an infinite family of pairwise weakly convex orthogonal convex types, then the theory has the maximal number of countable models (Corollary 4).

Weakly o-minimal theories have monotonicity [10] and strong monotonicity [11]. The last is used to confirm that weakly o-minimal theories have monotonic non-orthogonality on equivalence classes (Proposition 3). Thereby, Theorem 4 guarantees the mentioned result for weakly o-minimal theories of an arbitrary convexity rank (Corollary 5).

In a 1996 preprint, B. Baizhanov introduced the notions of a convex to the right (left) formula, a quasisolitary and a social type. These notions were later included in the works [12,13], which are partially based on this preprint. Theorem 8 and Corollary 7 establish the connection between these notions and the weak and almost orthogonality of 1-types in weakly o-minimal theories.

In Ref. [14] an example of a weakly o-minimal theory with countable number of countable non-isomorphic models was given. Theorem 9 provides a condition under which a weakly o-minimal theory has an infinite number of countable models.

2. Preliminaries

Let $\mathfrak{M}=\langle M;=,<,\dots \rangle$ be a linearly ordered structure. Although most of the results hold for the general case, we assume the structure $\mathfrak{M}$ is densely ordered [15,16].

For subsets A and B of M, we use the following notations:

$$A^{+}:=\{\gamma \in M\mid \mathfrak{M}\vDash a<\gamma \mathrm{for} \mathrm{all}a\in A\};$$

$$A^{-}:=\{\gamma \in M\mid \mathfrak{M}\vDash \gamma <a \mathrm{for} \mathrm{all}a\in A\}.$$

We write $A<B$ if for all $a\in A$, $b\in B$, $\mathfrak{M}\vDash a<b$. We write $a<B$ ($A<b$) if $\left\{a\right\}<B$ ($A<\left\{b\right\}$). If sets A and B are C-definable ($C\subseteq M$), then $A^{+},A^{-}$ and $A<B$ are C-definable as well.

Definition 4.

A subset A of a linearly ordered structure $\mathfrak{M}$ is said to be convex if $a<c<b$ implies $c\in A$ for all $a,b\in A$ and all $c\in M$.

Definition 5.

A formula $\phi (x,\overline{y},\overline{a})$ is a convex formula, if for every $\overline{b}\in M$ the set $\phi (\mathfrak{M},\overline{b},\overline{a})$ is convex in every model of $Th\left(\mathfrak{M}\right)$ containing $\overline{b}$ and $\overline{a}$.

In general, if a formula $\phi (y_1,y_2,\dots ,y_n,\overline{a})$ defines a convex set when any assignment is given to all the variables but $y_i$, we say that $\phi (y_1,y_2,\dots ,y_n,\overline{a})$ is convex in $y_i$.

Definition 6

([17]). (1) The convex closure of a formula $\phi (x,\overline{a})$ is the following formula:

${\phi }^c(x,\overline{a}):=\exists y_1\exists y_2\left(\phi (y_1,\overline{a})\wedge \phi (y_2,\overline{a})\wedge (y_1\le x\le y_2)\right)$.

(2) The convex closure of a type $p\left(x\right)\in S_1\left(A\right)$ is the following type:

$p^c\left(x\right):=\{{\phi }^c(x,\overline{a})\mid \phi (x,\overline{a})\in p\}$.

Similarly, denote $t{p}^c(\alpha /A):=\{{\phi }^c(x,\overline{a})\mid \phi (x,\overline{a})\in tp(\alpha /A)\}$. We define $S_p^c\left(A\right):=\{q\in S_1\left(A\right)\mid q^c=p^c\}$, and $S^c\left(A\right):=\left\{p^c\right|p\in S_1\left(A\right)\}$. Note that the type $p^c$ is not complete, and has a unique completion to a type from $S_1\left(A\right)$ if and only if $|S_p^c\left(A\right)|=1$. We call the type $p^c$ a convex type.

In o-minimal, weakly o-minimal, and quite o-minimal theories, the set of all realizations of any complete 1-type p is convex, that is, $|S_p^c\left(A\right)|=1$, or, equivalently, $p\left(\mathfrak{M}\right)=p^c\left(\mathfrak{M}\right)$.

Each convex type is of one of the following kinds.

Definition 7.

Let T be a linearly ordered theory, and $p^c\in S^c\left(A\right)$ be a convex 1-type, where A is a subset of an ${\left|A\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$.

(1) The type $p^c$ is isolated if there exists a convex formula $\phi (x,\overline{a})$, $\overline{a}\in A$, such that $p^c\left(\mathfrak{M}\right)=\phi (\mathfrak{M},\overline{a})$.

(2) The type $p^c$ is quasirational to the right (left) if there exists a convex formula $U(x,\overline{a})$, $a\in A$, such that for each $\alpha \in p\left(\mathfrak{M}\right)$ and each $\beta \ge \alpha$ ($\beta \le \alpha$)

$\beta \in p^c\left(\mathfrak{M}\right)$ if and only if $\beta \in U(\mathfrak{M},\alpha )$.

(3) The type $p^c$ is irrational if it is neither isolated nor quasirational.

Definition 8

([9]). Let $p^c,q^c\in S^c\left(A\right)$. We say that $p^c$ is not weakly convex-orthogonal to $q^c$, $p^c \cancel{\perp }{}_{ }{}^{cw }{q}^c_{ }^{ }$, if there exists a convex formula $H(x,y,\overline{a})$, $\overline{a}\in A$, such that $H(\mathfrak{M},\alpha ,\overline{a})\cap q^c\left(\mathfrak{M}\right)\ne \varnothing$ and $\neg H(\mathfrak{M},\alpha ,\overline{a})\cap q^c\left(\mathfrak{M}\right)\ne \varnothing$ for all (equivalently, for some) $\alpha \in p^c\left(\mathfrak{M}\right)$.

Definition 9.

Let $1\le n<\omega$, $\Gamma =\{p_1^c,p_2^c,\dots ,p_n^c\}\subseteq S^c\left(A\right)$. We say that the family Γ is not weakly convex orthogonal if there is an A-formula $H\left(\overline{y}\right)$ ($ln\left(\overline{y}\right)=n$) such that there exists $1\le i\le n$ such that $H\left(\overline{y}\right)$ is convex in $y_i$ and for some ${\alpha }_1\in p_1^c\left(\mathfrak{M}\right),{\alpha }_2\in p_2^c\left(\mathfrak{M}\right),\dots ,{\alpha }_{i-1}\in p_{i-1}^c\left(\mathfrak{M}\right),p_{i+1}^c\left(\mathfrak{M}\right),\dots ,{\alpha }_n\in p_n^c\left(\mathfrak{M}\right)$,

$H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)\cap p_i^c\left(\mathfrak{M}\right)\ne \varnothing$ and $\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)\cap p_i^c\left(\mathfrak{M}\right)\ne \varnothing$.

Then, the family $\Gamma$ is weakly convex orthogonal if and only if for each $1\le i\le n$ and each ${\overline{\alpha }}_i=({\alpha }_1,\dots ,{\alpha }_{i-1},{\alpha }_{i+1}\dots ,{\alpha }_n)$, where ${\alpha }_j\in p_j^c\left(\mathfrak{M}\right)$ for all $j\ne i$,

$t{p}^c({\alpha }_i/{\overline{\alpha }}_i)=t{p}^c({\alpha }_i^{\prime }/{\overline{\alpha }}_i)$ for all ${\alpha }_i,{\alpha }_i^{\prime }\in p_i^c\left(\mathfrak{M}\right)$.

Definition 10.

Let $A,B\subseteq M$, $H(x,y)$ be an A-formula, and $E(x,y)$ be an (parametrically definable) equivalence relation with convex classes.

(1) The formula $H(x,y)$ monotonically increases (decreases) on B, if for all $b_1,b_2\in B$, with $b_1<b_2$, the following holds:

$$H{(\mathfrak{M},b_1)}^{-}\subseteq H{(\mathfrak{M},b_2)}^{-} and H{(\mathfrak{M},b_2)}^{+}\subseteq H{(\mathfrak{M},b_1)}^{+}$$

$$(H{(\mathfrak{M},b_2)}^{-}\subseteq H{(\mathfrak{M},b_1)}^{-} and H{(\mathfrak{M},b_1)}^{+}\subseteq H{(\mathfrak{M},b_2)}^{+}),$$

and for some distinct $b_3,b_4\in B$, $H(\mathfrak{M},b_3)\ne H(\mathfrak{M},b_4)$.

(2) The formula $H(x,y)$ monotonically increases (decreases) on $B/E$, if for all $b_1,b_2\in B$, with $b_1<b_2$ and $\mathfrak{M}\vDash \neg E(b_1,b_2)$, the following holds:

$$H{(\mathfrak{M},b_1)}^{-}\subseteq H{(\mathfrak{M},b_2)}^{-} and H{(\mathfrak{M},b_2)}^{+}\subseteq H{(\mathfrak{M},b_1)}^{+}$$

$$(H{(\mathfrak{M},b_2)}^{-}\subseteq H{(\mathfrak{M},b_1)}^{-} and H{(\mathfrak{M},b_1)}^{+}\subseteq H{(\mathfrak{M},b_2)}^{+}),$$

and for some distinct $b_3,b_4\in B$, $\mathfrak{M}\vDash \neg E(b_3,b_4)$ and $H(\mathfrak{M},b_3)\ne H(\mathfrak{M},b_4)$.

Let $H(y_1,y_2,\dots ,y_n)$ be an A-formula, $i,j\le n$ be arbitrary and distinct, and $\overline{b}\in \mathfrak{M}$ be an $(n-2)$-tuple. Then, we let $H_{i,j}^{\overline{b}}(y_j,y_i)$ be the result of the substitution of $\overline{b}$ for all the parameters of H but $y_i$ and $y_j$. More precisely, if, without loss of generality, $i<j$, and $\overline{b}=(b_1,b_2,\dots ,b_{i-1},b_{i+1},\dots ,b_{j-1},b_{j+1},\dots ,b_n)$, then define

$$H_{i,j}^{\overline{b}}(x_j,x_i):=H(b_1,\dots ,b_{i-1},y_i,b_{i+1},\dots ,b_{j-1},y_j,b_{j+1},\dots ,b_n).$$

Then, $H_{i,j}^{\overline{b}}(x_j,x_i)$ is an $(A\cup \overline{b})$-definable 2-formula.

Definition 11.

(1) An A-formula $H(y_1,y_2,\dots ,y_n)$ monotonically increases (decreases) in $y_j$ with respect to $y_i$ ($1\le i\ne j\le n$) on $B_1,\dots ,B_{j-1},B_{j+1},\dots ,B_n\subseteq M$, if the formula $H_{i,j}^{\overline{b}}(x_j,x_i)$ monotonically increases (decreases) on $B_i$ for all assignments $\overline{b}={\left(b_k\right)}_{k\ne i,j}$, $b_k\in B_k$, $k\in \{1,\dots ,n\}\setminus \{i,j\}$.

(2) An A-formula $H(x,y_1,y_2,\dots ,y_n)$ monotonically increases (decreases) on equivalence classes in $y_j$ with respect to $y_i$ ($1\le i\ne j\le n$) on $B_1,\dots ,B_{j-1},B_{j+1},\dots ,B_n\subseteq M$, if the formula $H_{i,j}^{\overline{b}}(x_j,x_i)$ monotonically increases (decreases) on $B_i/E_i(x,y,\overline{b})$ for some equivalence relation $E_i(x,y,\overline{z})$ and all assignments $\overline{b}={\left(b_k\right)}_{k\ne i,j}$, $b_k\in B_k$, $k\in \{1,\dots ,n\}\setminus \{i,j\}$.

We say that a formula is monotonic, if it either monotonically increases or monotonically decreases.

Definition 12.

Let T be a linearly ordered theory.

(1) The theory T has monotonic non-orthogonality if for each $n<\omega$ and each non-weakly convex orthogonal family $\Gamma =\{p_1^c,p_2^c,\dots ,p_n^c\}\in S^c\left(A\right)$ of non-isolated convex types such that each proper subset ${\Gamma }^{\prime }⊊\Gamma$ is weakly convex orthogonal, where $A\subseteq \mathfrak{M}$ is finite, and $\mathfrak{M}\vDash T$ is ${\left|A\right|}^{+}$-saturated, there exists an A-definable formula $H(y_1,y_2,\dots ,y_n)$ as in Definition 9 such that $H(y_1,y_2,\dots ,y_n)$ is convex in $y_j$ and strictly monotonic in $y_j$ with respect to $y_i$ on $p_1^c\left(\mathfrak{M}\right)$, $p_2^c\left(\mathfrak{M}\right),\dots ,p_{j-1}^c\left(\mathfrak{M}\right)$, $p_{j+1}^c\left(\mathfrak{M}\right),\dots$, $p_n^c\left(\mathfrak{M}\right)$ for each distinct $1\le i,j\le n$.

(2) The theory T has monotonic non-orthogonality on equivalence classes if for each $n<\omega$ and each finite non-weakly convex orthogonal family $\Gamma =\{p_1^c,p_2^c,\dots ,p_n^c\}\in S^c\left(A\right)$ of non-isolated convex types such that each proper subset ${\Gamma }^{\prime }⊊\Gamma$ is weakly convex orthogonal, where $A\subseteq \mathfrak{M}$ is finite, and $\mathfrak{M}\vDash T$ is ${\left|A\right|}^{+}$-saturated, there exists an A-definable formula $H(y_1,y_2,\dots ,y_n)$ as in Definition 9 such that it is convex in $y_j$ and is strictly monotonic on equivalence classes in $y_j$ with respect to $y_i$ on $p_1^c\left(\mathfrak{M}\right)$, $p_2^c\left(\mathfrak{M}\right),\dots ,p_{j-1}^c\left(\mathfrak{M}\right)$, $p_{j+1}^c\left(\mathfrak{M}\right),\dots$, $p_n^c\left(\mathfrak{M}\right)$ for each $i,j$ with $1\le i\ne j\le n$.

Each theory with monotonic non-orthogonality has monotonic non-orthogonality on equivalence classes by the trivial equivalence relation $x=y$. On the other hand, a theory with monotonic non-orthogonality on equivalence classes does not necessarily have monotonic non-orthogonality. But such theories retain some properties of theories with monotonic non-orthogonality by considering corresponding equivalence classes instead of the elements themselves. More precisely, if T has monotonic non-orthogonality on equivalence classes and $p^c,q^c$ and H are as in Definition 8, and H monotonically increases (decreases) on $p^c\left(\mathfrak{M}\right)/E$, denote $H^E(x,y):=\exists z\left(E(z,y)\wedge H(x,z)\right)$. Then, the formula $H^{E_i}$ is constant on each $E(\mathfrak{M},\beta )$, $\beta \in p^c\left(\mathfrak{M}\right)$, and monotonically increases (decreases) on $p^c\left(\mathfrak{M}\right)$. Analogically, the formula $H^{E_i}(y_1,\dots ,y_n):=\exists z_i\left(E_i(z_i,y_i)\wedge H(y_1,\dots ,y_{i-1},z_i,y_{i+1},\dots ,y_n)\right)$ can be taken for a formula H from Definition 9 which is monotonic on equivalence classes in $y_j$ with respect to $y_i$ on $p_1^c\left(\mathfrak{M}\right)$, $p_2^c\left(\mathfrak{M}\right),\dots ,p_{j-1}^c\left(\mathfrak{M}\right)$, $p_{j+1}^c\left(\mathfrak{M}\right),\dots$, $p_n^c\left(\mathfrak{M}\right)$, where $E_i$ is the corresponding equivalence relation.

Therefore, below, without loss of generality, it is enough to give proofs for theories with monotonic non-orthogonality rather than those with monotonic non-orthogonality on equivalence classes.

Example 1.

Let $\mathfrak{M}={\langle \mathbb{Q}+(\mathbb{Q}+\sqrt{2});=,<,P,U,a_i,b_i,\rangle }_{i<\omega }$, where < is the standard dense linear order without endpoints, P is a unary predicate and U is a binary predicate. Let $a_{i_1}<a_{i_2}<b_{j_2}<b_{j_1}$ for all natural $i_1<i_2$, $j_1<j_2$, ${lim}_{i\to \infty }{a}_i={lim}_{j\to \infty }{b}_j=e$, and ${\{a_i,b_i\}}_{i<\omega }\subset \mathbb{Q}=P\left(\mathfrak{M}\right)<\neg P\left(\mathfrak{M}\right)=(\mathbb{Q}+\sqrt{2})$. Define $\mathfrak{M}\vDash U(a,b)$ if and only if $\mathfrak{M}\vDash P\left(a\right)\wedge \ne P\left(b\right)$ and $\mathbb{R}\vDash a<b$.

Define $p\left(x\right):=\{a_i<x<b_j|i,j\in \omega \}$, and $q\left(y\right):=\{U(a_i,y)\wedge \neg U(b_j,y)|i,j\in \omega \}$. The types p and q are convex, and are the only two distinct non-isolated types over the empty set, and $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$. There are no other distinct non-isolated types over finite sets. Then, the formula $U(x,y)$ monotonically increases on $p\left(\mathfrak{N}\right)$ for each $\mathfrak{N}\in Mod\left(\mathfrak{M}\right)$. Therefore, $Th\left(\mathfrak{M}\right)$ has monotonic non-orthogonality.

Example 2.

Let $C={\left\{c_i\right\}}_{i<\omega }\subset \mathbb{R}\setminus \mathbb{Q}$ be such that for each $i<\omega$ $c_i\notin acl((C\setminus \left\{c_i\right\})\cup \{\sqrt{2},e\})$. For each $i<\omega$ denote ${\mathbb{Q}}_i:=\mathbb{Q}+c_i$.

Let $\mathfrak{M}={\langle M;=,<,P_i,S,U,a_i,b_i\rangle }_{i<\omega }$, where $M=\left(\mathbb{Q}\cup {\displaystyle \bigcup _{i<\omega }}{\mathbb{Q}}_i\right)+\left({\mathbb{Q}}_{\sqrt{2}}\times \mathbb{Q}\right)$, $P_i$ and S are unary predicates and U is a binary predicate. Let $S\left(\mathfrak{M}\right)={\mathbb{Q}}_{\sqrt{2}}\times \mathbb{Q}$, $\neg S\left(\mathfrak{M}\right)=\mathbb{Q}\cup {\displaystyle \bigcup _{i<\omega }}{\mathbb{Q}}_i$, $\neg S\left(\mathfrak{M}\right)<S\left(\mathfrak{M}\right)$, $P\left(\mathfrak{M}\right)=\mathbb{Q}\subset \neg S\left(\mathfrak{M}\right)$, and $P_i\left(\mathfrak{M}\right)={\mathbb{Q}}_i\subset \neg S\left(\mathfrak{M}\right)$. Let < be the standard relation of linear order on $\neg S\left(\mathfrak{M}\right)$, and the lexicographic order on $S\left(\mathfrak{M}\right)$. That is, for each $r_1,r_2\in {\mathbb{Q}}_{\sqrt{2}},q_1,q_2\in \mathbb{Q}$, let $(r_1,q_1)<(r_2,q_2)$ if and only if on $\mathbb{R}$ either $r_1<r_2$, or $r_1=r_2$ and $q_1<q_2$.

Analogically to the previous example, let $a_{i_1}<a_{i_2}<b_{j_2}<b_{j_1}$ for all natural $i_1<i_2$, $j_1<j_2$, ${lim}_{i\to \infty }{a}_i={lim}_{j\to \infty }{b}_j=e$, and ${\{a_i,b_i\}}_{i<\omega }\subset \mathbb{Q}\subset \neg S\left(\mathfrak{M}\right)$.

Let $\mathfrak{M}\vDash \forall x\forall y\left(U(x,y)\to (\neg S(x)\wedge S(y)\right))$, and for all $a\in \neg S\left(\mathfrak{M}\right),b=(r+\sqrt{2},q)\in S\left(\mathfrak{M}\right)$, let $\mathfrak{M}\vDash U(a,b)$ if and only if $\mathbb{R}\vDash a<r+\sqrt{2}$.

For $k\in \{0,1,2,\dots \}$, define $p_k\left(x\right):=\{a_i<x<b_j|i,j<\omega \}\cup \left\{P_k\left(x\right)\right\}$. All the $p_k$ are the complete non-isolated mutually dense types. Moreover, for each $\mathfrak{N}\in Mod\left(Th\right(\mathfrak{M}\left)\right)$ and each distinct $k_1$ and $k_2$, $p_{k_1}^c\left(\mathfrak{N}\right)=p_{k_2}^c\left(\mathfrak{N}\right)$, and this convex closure is equal to the set of all realizations of the incomplete type $\{a_i<x<b_j|i,j<\omega \}$ in $\mathfrak{N}$. Let $q\left(y\right):=\{U(a_i,y)\wedge \neg U(b_j,y)|i,j\in \omega \}$. Then, $q\left(\mathfrak{N}\right)=q^c\left(\mathfrak{N}\right)$, and for each $k\ge 0$ $p_k \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$ and $p_k^c \cancel{\perp }{}_{ }{}^{cw }{q}^c_{ }^{ }$.

Let $E(y_1,y_2):=\forall x\left(\neg S\left(x\right)\to \left(U(x,y_1)\leftrightarrow U(x,y_2)\right)\right)$. Then, E is an ∅-definable equivalence relation with convex classes on $S\left(\mathfrak{M}\right)$. Then, $\mathfrak{M}\vDash E\left((r_1+\sqrt{2},q_1),(r_2+\sqrt{2},q_2)\right)$ if and only if $r_1=r_2$. Then, the formula U is monotonic on $p^c\left(\mathfrak{N}\right)$ and strictly monotonic on equivalence classes on $q^c\left(\mathfrak{N}\right)$. Therefore, this is an example of a theory with monotonic non-orthogonality on equivalence classes but $Th\left(\mathfrak{M}\right)$ is not a theory with monotonic non-orthogonality.

In theories with monotonicity, the definition of weak convex orthogonality of 1-types coincides with the definition of weak convex orthogonality from [9].

Theorem 2

([9]). Let T be a countable linearly ordered theory with monotonic non-orthogonality, $A\subset M$, $\mathfrak{M}\vDash T$ be ${\left|A\right|}^{+}$-saturated, and $p,q\in S_1\left(A\right)$ be non-isolated types. Then, $p^c \cancel{\perp }{}_{ }{}^{cw }{q}^c_{ }^{ }$ if and only if $q^c \cancel{\perp }{}_{ }{}^{cw }{p}^c_{ }^{ }$.

Corollary 1.

Let T be a countable linearly ordered theory with monotonic non-orthogonality on equivalence classes, $A\subset M$, $\mathfrak{M}\vDash T$ be ${\left|A\right|}^{+}$-saturated, and $p,q\in S_1\left(A\right)$ be non-isolated types. Then, $p^c \cancel{\perp }{}_{ }{}^{cw }{q}^c_{ }^{ }$ if and only if $q^c \cancel{\perp }{}_{ }{}^{cw }{p}^c_{ }^{ }$.

Corollary 2.

Let T be a countable linearly ordered theory with monotonic non-orthogonality on equivalence classes, $A\subset M$, $\mathfrak{M}\vDash T$ be ${\left|A\right|}^{+}$-saturated. Then, $p^c \cancel{\perp }{}_{ }{}^{cw }{q}^c_{ }^{ }$ is an equivalence relation on $S^c\left(A\right)$.

Proof of Corollary 2.

As noted above, without loss of generality, assume that T has monotonic non-orthogonality.

From the definition it follows that $p^c \cancel{\perp }{}_{ }{}^{cw }{p}^c_{ }^{ }$ for each $p^c\in S^c\left(A\right)$. Theorem 2 proves symmetry.

Let $p^c,q^c,r^c\in S^c\left(A\right)$ be such that $p^c \cancel{\perp }{}_{ }{}^{cw }{q}^c_{ }^{ }$ and $q^c \cancel{\perp }{}_{ }{}^{cw }{r}^c_{ }^{ }$. Let $H_1(x,y)$ and $H_2(x,y)$ be convex A-formulas guaranteeing these two non-weak orthogonalities, respectively. We can transform these formulas so that for all $\alpha \in p^c\left(\mathfrak{M}\right)$, $\beta \in q^c\left(\mathfrak{M}\right)$, $(H_1(\mathfrak{M},\alpha )={\left(H_1{(\mathfrak{M},\alpha )}^{+}\right)}^{-}$, $(H_2(\mathfrak{M},\beta )={\left(H_1{(\mathfrak{M},\beta )}^{+}\right)}^{-}$, and the sets $H_1(\mathfrak{M},\alpha )\cap q^c\left(\mathfrak{M}\right)$, $\neg H_1(\mathfrak{M},\alpha )\cap q^c\left(\mathfrak{M}\right)$, $H_2(\mathfrak{M},\beta )\cap r^c\left(\mathfrak{M}\right)$ and $\neg H_2(\mathfrak{M},\beta )\cap r^c\left(\mathfrak{M}\right)$ are all non-empty.

By compactness, there is a formula $K\left(z\right)\in q^c\left(z\right)$ such that the formula $H_2$ is a convex monotonic formula on $K\left(\mathfrak{M}\right)$ splitting $r^c\left(\mathfrak{M}\right)$ and with $(H_2(\mathfrak{M},\beta )={\left(H_2{(\mathfrak{M},\beta )}^{+}\right)}^{-}$ for each $\beta \in K\left(\mathfrak{M}\right)$. Then, the formula $H_3(x,y):=\exists z\left(K\left(z\right)\wedge H_1(z,y)\wedge H_2(x,z)\right)$ guarantees that $p^c{\perp }^{cw}{r}^c$, thus proving transitivity. □

3. Theories with Monotonic Non-Orthogonality

Theorem 3.

Let T be a countable theory, $p\in S_1\left(A\right)$, A be a finite subset of a countably saturated model $\mathfrak{M}\vDash T$. If p is non-isolated, then there exists a countable family of 1-A-formulas ${\left\{{\phi }_i\left(x\right)\right\}}_{i<\omega }$ such that ${\displaystyle \bigcap _{i<\omega }}{\phi }_i\left(\mathfrak{M}\right)=p\left(\mathfrak{M}\right)$ and ${\phi }_{i+1}\left(\mathfrak{M}\right)⊊{\phi }_i\left(\mathfrak{M}\right)$ for all $i<\omega$.

Proof of Theorem 3.

Let ${\left\{{\theta }_i\right\}}_{i<\omega }$ be an enumeration of all formulas of p. Denote ${\phi }_1:={\theta }_1$. Let $i>1$ be the smallest index such that ${\theta }_1\left(\mathfrak{M}\right)⊊{\theta }_i\left(\mathfrak{M}\right)⊊p\left(\mathfrak{M}\right)$. Such a formula exists since the type p is non-isolated. Denote ${\phi }_2:={\theta }_i$. Continuing this construction by induction, we obtain a countable family ${\left\{{\phi }_i\right\}}_{i<\omega }$ of nested formulas. By construction, ${\displaystyle \bigcap _{i<\omega }}{\phi }_i\left(\mathfrak{M}\right)\supseteq p\left(\mathfrak{M}\right)$.

Note that for each $i<\omega$ there is $j<\omega$ such that ${\phi }_j\left(\mathfrak{M}\right)\subseteq {\theta }_i\left(\mathfrak{M}\right)$. This is true because on step i either ${\theta }_i$ is chosen as ${\phi }_i$, or a smaller formula has already been chosen before, that is, ${\phi }_{i-1}\left(\mathfrak{M}\right)⊊{\theta }_i\left(\mathfrak{M}\right)$.

Towards a contradiction, suppose that there is $c\in {\displaystyle \bigcap _{i<\omega }}{\phi }_i\left(\mathfrak{M}\right)\setminus p\left(\mathfrak{M}\right)$. Then, there is $\phi \in p$ such that $c\notin \phi \left(\mathfrak{M}\right)$. For some $k<\omega$, $\phi ={\theta }_k$. Then, there is $k^{\prime }\ge k$ such that ${\phi }_{k^{\prime }}\left(\mathfrak{M}\right)\subseteq \phi \left(\mathfrak{M}\right)={\theta }_k\left(\mathfrak{M}\right)$. This is a contradiction with the choice of c. Therefore, ${\displaystyle \bigcap _{i<\omega }}{\phi }_i\left(\mathfrak{M}\right)=p\left(\mathfrak{M}\right)$. □

The proof of Theorem 3 can be repeated for convex types by considering only convex formulas. Then, the following holds.

Corollary 3.

Let T be a countable linearly ordered theory, $p\in S_1\left(A\right)$, A be a finite subset of a countably saturated model $\mathfrak{M}\vDash T$. If $p^c$ is non-isolated, then there exists a countable family of convex 1-A-formulas ${\left\{{\phi }_i\left(x\right)\right\}}_{i<\omega }$ such that ${\displaystyle \bigcap _{i<\omega }}{\phi }_i\left(\mathfrak{M}\right)=p^c\left(\mathfrak{M}\right)$ and ${\phi }_{i+1}\left(\mathfrak{M}\right)⊊{\phi }_i\left(\mathfrak{M}\right)$ for all $i<\omega$.

Theorem 4.

Let T be a small theory with monotonic non-orthogonality on equivalence classes. Let Γ be a pairwise weakly convex orthogonal set of non-isolated convex types from $S^c\left(A\right)$ for a finite set A. Then, Γ is weakly convex orthogonal.

As previously, without loss of generality, assume that T has monotonic non-orthogonality.

Proof of Theorem 4.

Analogically to L. Mayer’s proof (Theorem 2.4 from [2]), it is sufficient to confirm the following claim.

Claim 1.

Let Γ be a finite weakly convex orthogonal set of non-isolated convex 1-types from $S^c\left(A\right)$. If $q^c\in S^c\left(A\right)$ is such that $\Gamma \cup q^c$ is not weakly convex orthogonal, then there is $p^c\in \Gamma$ such that $q^c \cancel{\perp }{}_{ }{}^{cw }{p}^c_{ }^{ }$.

The proof is by induction on $n:=|\Gamma |$. The case when $n=2$ is trivial.

Let $n>2$. For induction, assume that the claim holds for all $m<n$. Let $\Gamma =\{p_1^c,p_2^c,\dots ,p_n^c\}$ be a weakly orthogonal set of non-isolated 1-types from $S_1^c\left(T\right)$. Since all $p_i^c$ are non-isolated, without loss of generality, we can assume that each $p_i^c{\left(\mathfrak{M}\right)}^{-}$ is non-definable. That is, for each $i<\omega$, $p_i$ is either irrational or quasirational to the right.

Let $H(z,x_1,\dots ,x_n)$ be an A-definable formula such that for some ${\alpha }_1\in p_1^c\left(\mathfrak{M}\right)$, ${\alpha }_2\in p_2^c\left(\mathfrak{M}\right)$, …, ${\alpha }_n\in p_n^c\left(\mathfrak{M}\right)$, the set $H(\mathfrak{M},{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$ is convex, there are ${\gamma }_1,{\gamma }_2\in q^c\left(\mathfrak{M}\right)$, ${\gamma }_1\in (q^c\left(\mathfrak{M}\right)\cap H(\mathfrak{M},{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n))<{\gamma }_2$, and $q^c{\left(\mathfrak{M}\right)}^{-}\subseteq H(\mathfrak{M},{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$. Let for each i, $1\le i\le n$, $P_i\in p_i$ be such that $H(z,x_1,\dots ,x_n)$ is monotonic in z on $P_1\left(\mathfrak{M}\right),\dots P_n\left(\mathfrak{M}\right)$ with respect to each $x_i$, $1\le i\le n$, without loss of generality, let H be strictly increasing in z for each i.

Let ${\phi }_1,{\phi }_2$ be such convex A-formulas that $P_1{\left(\mathfrak{M}\right)}^{-}\subseteq {\phi }_1\left(\mathfrak{M}\right)⊊{\phi }_2\left(\mathfrak{M}\right)$, ${\phi }_1{\left(\mathfrak{M}\right)}^{+}⊋{\phi }_2{\left(\mathfrak{M}\right)}^{+}⊋p_1^c\left(\mathfrak{M}\right)$. Without loss of generality, let ${\phi }_1{\left(\mathfrak{M}\right)}^{-}={\phi }_2{\left(\mathfrak{M}\right)}^{-}=\varnothing$. Such two formulas exist because of Corollary 3. For each $i\in \{1,2\}$, denote

$$R_i(z,{\alpha }_2,\dots ,{\alpha }_n):=\exists x({\phi }_i\left(x\right)\wedge P_1\left(x\right)\wedge H(z,x,{\alpha }_2,\dots ,{\alpha }_n)).$$

Monotonicity of H implies $R_1(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)⊊R_2(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)<q\left(\mathfrak{M}\right)$. Let ${\left\{{\psi }_j\right\}}_{j<\omega }$ be the sequence of convex formulas defining $p_2^c\left(\mathfrak{M}\right)$ as in Corollary 3. Denote

$$K_j(z,{\alpha }_3,\dots ,{\alpha }_n):=\exists x\exists y(H(z,x,y,{\alpha }_3,\dots {\alpha }_n)\wedge {\phi }_2\left(x\right)\wedge {\psi }_j{\left(y\right)}^{-}\wedge P_2\left(y\right))$$

While j approaches infinity, $K_j(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)$ approaches $R_2(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)$. Monotonicity and compactness imply the existence of $t<\omega$ such that $R_1(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)⊊K_t(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊R_2(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)$.

Denote ${\phi }^{\prime }(x,{\alpha }_2,\dots ,{\alpha }_n):=\exists z(K_t(z,{\alpha }_3,\dots ,{\alpha }_n)\wedge H(z,x,{\alpha }_2,\dots ,{\alpha }_n)\wedge {\phi }_2\left(x\right))$. Then, $(P_1\left(\mathfrak{M}\right)\cap {\phi }_1\left(\mathfrak{M}\right))\subseteq (P_1\left(\mathfrak{M}\right)\cap {\phi }^{\prime }\left(\mathfrak{M}\right))\subseteq (P_1\left(\mathfrak{M}\right)\cap {\phi }_2\left(\mathfrak{M}\right))$. The pairs of formulas ${\phi }_1,{\phi }^{\prime }$ and ${\phi }^{\prime },{\phi }_2$ behave analogically to the pair ${\phi }_1,{\phi }_2$. Then, we can apply the previous consideration to construct two convex formulas $K^0$ and $K^1$ such that $R_1(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)⊊K^0(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K_t(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K^1(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊R_2(\mathfrak{M},{\alpha }_2,\dots ,{\alpha }_n)$.

For each $i\in \{0,1\}$, and later, for each finite sequence of zeroes and ones, denote ${\phi }^i(x,{\alpha }_2,\dots ,{\alpha }_n):=\exists z(K^i(z,{\alpha }_3,\dots ,{\alpha }_n)\wedge H(z,x,{\alpha }_2,\dots ,{\alpha }_n)\wedge {\phi }_2\left(x\right))$. Then, $(P_1\left(\mathfrak{M}\right)\cap {\phi }_1\left(\mathfrak{M}\right))\subseteq (P_1\left(\mathfrak{M}\right)\cap {\phi }^0\left(\mathfrak{M}\right))\subseteq (P_1\left(\mathfrak{M}\right)\cap {\phi }^{\prime }\left(\mathfrak{M}\right))\subseteq (P_1\left(\mathfrak{M}\right)\cap {\phi }^1\left(\mathfrak{M}\right))\subseteq (P_1\left(\mathfrak{M}\right)\cap {\phi }_2\left(\mathfrak{M}\right))$.

Using the new pairs of formulas, we can continue dividing the definable sets: ${\phi }_1,{\phi }^0$ produce $K^{00}$, ${\phi }^0,{\phi }^{\prime }$ produce $K^{01}$, ${\phi }^{\prime },{\phi }^1$ produce $K^{10}$, and ${\phi }^1,{\phi }_2$ produce $K^{11}$.

Repeating this construction, for each finite sequence $\tau$ of zeroes and ones we obtain a convex formula $K^{\tau }$ such that $K^{\tau 00}(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K^{\tau 0}(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K^{\tau }(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K^{\tau 10}(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K^{\tau 1}(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)⊊K^{\tau 11}(\mathfrak{M},{\alpha }_3,\dots ,{\alpha }_n)$. This gives us a continuum of 1-types over $\{{\alpha }_3,\dots ,{\alpha }_n\}$, which contradicts the theory $T\cup tp({\alpha }_3,\dots ,{\alpha }_n)$ being small. □

Theorem 5

([18]). Let $\mathfrak{M}\vDash T$ be a model of a small countable complete theory T. Let $B\subseteq M$ be countable. Then, there exists a countable model $\mathfrak{A}\vDash T$ such that $B\subseteq A$ and for every $\overline{a}\in A$ there is $\overline{b}\in B$, such that $tp(\overline{a}/\overline{b})$ is isolated.

Corollary 4.

Let T be a countable complete linearly ordered theory with monotonic non-orthogonality on equivalence classes, and let Γ be an infinite family of pairwise weakly convex orthogonal convex 1-types from $S^c\left(A\right)$ for a finite set A. Then, $I(T,{\aleph }_0)=2^{{\aleph }_0}$.

Proof Corollary 4.

Without loss of generality, let the theory T have monotonic non-orthogonality. It is enough to only consider the case when T is small. Since otherwise the theory T has $2^{{\aleph }_0}$ countable non-isomorphic models. Extend the language of T to $\mathcal{L}\left(A\right)$, where L is the language of T, and work in the theory $T\cup tp\left(\overline{a}\right)$, where $\overline{a}$ is some enumeration of the set A. Let $\mathfrak{N}$ be an ${\aleph }_1$-saturated elementary extension of $\mathfrak{M}$.

Fix an arbitrary enumeration $\Gamma ={\{p_1^c,p_2^c,\dots ,p_i^c,\dots \}}_{i<\omega }$. Let $\tau =\langle {\tau }_1,{\tau }_2,\dots ,{\tau }_i,\dots \rangle$, ${\tau }_i\in \{0,1\}$, be an arbitrary infinite sequence of zeroes and ones. Denote $I_{\tau }:=\{i<\omega |{\tau }_i=1\}$. Since $p_i^c\left(\mathfrak{N}\right)\ne \varnothing$, for each $i<\omega$, we can choose a special set of representatives $B_{\tau }={\left\{{\beta }_i\right\}}_{i\in I_{\tau }}\subseteq \mathfrak{N}$ of realizations of $p_i^c$’s such that for each $i\in I_{\tau }$ ${\beta }_i\in p_i^c\left(\mathfrak{N}\right)$. Let ${\mathfrak{A}}^{\tau }\vDash T$ be the model which exists by Theorem 5 applied to the set $B_{\tau }$.

Let $j<\omega$ be such that $j\notin I_{\tau }$. We claim that $p_j^c\left({\mathfrak{A}}^{\tau }\right)=\varnothing$. Towards a contradiction suppose that there is $\alpha \in p_j^c\left({\mathfrak{A}}^{\tau }\right)$. Then, by Theorem 5, there is a tuple $\overline{\beta }\in B_{\tau }$ such that $tp(\alpha /\overline{\beta })$ is isolated. Let $H(x,\overline{\beta })$ be the isolating formula of $tp(\alpha /\overline{\beta })$. Then, $H({\mathfrak{A}}^{\tau },\overline{\beta })=tp(\alpha /\overline{\beta })\left({\mathfrak{A}}^{\tau }\right)$. We have that $tp(\alpha /\overline{\beta })\subseteq tp\left(\alpha \right)\subseteq p_j^c\left({\mathfrak{A}}^{\tau }\right)$. Since the type $p_j^c$ is non-isolated, $H({\mathfrak{A}}^{\tau },\overline{\beta })\ne p_j^c\left({\mathfrak{A}}^{\tau }\right)$. Then, we obtain $H^c({\mathfrak{A}}^{\tau },\overline{\beta })⊊p_j^c\left({\mathfrak{A}}^{\tau }\right)$.

By construction of $B_{\tau }$, $\overline{\beta }=({\beta }_{i_1},{\beta }_{i_2},\dots ,{\beta }_{i_n})$ for some distinct $i_1,i_2,\dots i_n\in I_{\tau }$, where $n=ln\left(\overline{\beta }\right)$. Then, $H^c({\mathfrak{A}}^{\tau },\overline{\beta })$ guarantees that the family $\Gamma$ is not convex weakly orthogonal. But this is impossible since, by Theorem 4, the pairwise weakly convex orthogonal family $\Gamma$ should be weakly convex orthogonal.

Since we have obtained a contradiction, $p_j^c\left({\mathfrak{A}}^{\tau }\right)=\varnothing$. Because $j\notin I_{\tau }$ is arbitrary, ${\mathfrak{A}}^{\tau }$ realizes all $p_i$ with $i\in I_{\tau }$, and omits all $p_j$ with $j\notin I_{\tau }$. Therefore, for different infinite sequences ${\tau }_1$ and ${\tau }_2$ of zeroes and ones, the models ${\mathfrak{A}}^{{\tau }_1}$ and ${\mathfrak{A}}^{{\tau }_2}$ are non-isomorphic. This leads to continuum countable non-isomorphic models of the theory $T\cup tp\left(\overline{a}\right)$, as well as of the theory T. □

4. Weakly O-Minimal Theories

Let $\mathfrak{M}\vDash T$ be an ${\left|A\right|}^{+}$-saturated model of a weakly o-minimal theory T for $A\subseteq M$.

Definition 13

([12]). (1) An A-definable formula $\phi (x,y)$ is said to be p–preserving (p-stable) if there exist α, ${\gamma }_1$, ${\gamma }_2\in p\left(\mathfrak{M}\right)$ such that $p\left(\mathfrak{M}\right)\cap \left(\phi (\mathfrak{M},\alpha )\setminus \left\{\alpha \right\}\right)\ne \varnothing$ and ${\gamma }_1<\phi (\mathfrak{M},\alpha )<{\gamma }_2$.

(2) A p-preserving formula $\phi (x,y)$ is said to be convex to the right (left) on p if there exists α∈$p\left(\mathfrak{M}\right)$ such that $p\left(\mathfrak{M}\right)\cap \phi (\mathfrak{M},\alpha )$ is convex in the set $p\left(\mathfrak{M}\right)$ (that is, ${\gamma }_1,{\gamma }_2\in (p\left(\mathfrak{M}\right)\cap \phi (\mathfrak{M},\alpha ))$, $\delta \in p\left(\mathfrak{M}\right)$ and ${\gamma }_1<\delta <{\gamma }_2$ imply $\delta \in \phi (\mathfrak{M},\alpha )$), α is the left (right) endpoint of the set $\phi (\mathfrak{M},\alpha )$, and $\alpha \in \phi (\mathfrak{M},\alpha )$.

The family of all p-preserving convex to the right (left) A-formulas is denoted by $CRF\left(p\right)$ ($CLF\left(p\right)$). The set of all convex to the right (left) formulas on a 1-type is linearly ordered by inclusion. Such formulas play an important role in [6,7,8,19], as well as in other works by the authors.

Definition 14

([12]). (1) A type $p\in S_1\left(A\right)$ is quasisolitary if there exists a greatest convex to the right (equivalently, convex to the left) on p formula.

(2) A type $p\in S_1\left(A\right)$ is social if there exist no greatest convex to the right (equivalently, convex to the left) on p formula.

Definition 15

([12]). The type p is not almost orthogonal to the type q, $p \cancel{\perp }{}_{ }{}^{a }q_{ }^{ }$, if there exists an A-definable formula $H(x,y)$, such that for some $\alpha \in p\left(\mathfrak{N}\right)$ and ${\gamma }_1,{\gamma }_2\in q\left(\mathfrak{N}\right)$, ${\gamma }_1<H(\mathfrak{N},\alpha )<{\gamma }_2$ and $H(\mathfrak{N},\alpha )\cap q\left(\mathfrak{N}\right)\ne \varnothing$.

Note 1

([12]). Let T be a weakly o-minimal theory, let A be a subset of an ${\left|A\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$. Then, the relations ${\cancel{\perp }}^w$ and ${\cancel{\perp }}^a$ are equivalence relations on $S_1\left(A\right)$.

Recall that in weakly o-minimal theories, the set of all realizations of each complete 1-type is convex. Then, for such theories $p^c{\perp }^{cw}{q}^c$ if and only if $p{\perp }^{w}q$. Hence, we do not need to distinguish 1-types from their convex closures.

Proposition 1.

Let T be a weakly o-minimal theory, $A\subseteq \mathfrak{M}$, and $\mathfrak{M}\vDash T$ be ${\left|A\right|}^{+}$-saturated. Let $\Gamma =\{p_1,p_2,\dots ,p_n\}\in S\left(A\right)$ be a non-weakly orthogonal family of non-isolated 1-types such that each proper subset ${\Gamma }^{\prime }⊊\Gamma$ is weakly orthogonal. Then, there exists an A-definable formula $H(y_1,y_2,\dots ,y_n)$ that satisfies Definition 9 such that for all $1\le i\le n$, the formula $H(y_1,y_2,\dots ,y_n)$ is convex in $y_i$ and for all ${\alpha }_1\in p_1\left(\mathfrak{M}\right),{\alpha }_2\in p_2\left(\mathfrak{M}\right),\dots ,{\alpha }_{i-1}\in p_{i-1}\left(\mathfrak{M}\right),p_{i+1}\left(\mathfrak{M}\right),\dots ,{\alpha }_n\in p_n\left(\mathfrak{M}\right)$,

$$H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)\cap p_i\left(\mathfrak{M}\right)\ne \varnothing ,$$

$$\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)\cap p_i\left(\mathfrak{M}\right)\ne \varnothing , \mathit{and} \mathit{either}$$

$$\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)=H{({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)}^{+} \mathit{or}$$

$$\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)=H{({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)}^{-}$$

Proof of Proposition 1.

Non-weak orthogonality of $\Gamma$ implies that there are an n-A-formula $H_0$ and realizations ${\alpha }_1\in p_1\left(\mathfrak{M}\right),{\alpha }_2\in p_2\left(\mathfrak{M}\right),\dots ,{\alpha }_n\in p_n\left(\mathfrak{M}\right)$ such that $\mathfrak{M}\vDash H_0({\alpha }_1,{\alpha }_2,...,{\alpha }_n)$ and for some i, $1\le i\le n$, and some ${\alpha }_i^{\prime }\in p_i\left(\mathfrak{M}\right)$ we have $\mathfrak{M}\vDash \neg H_0({\alpha }_1,{\alpha }_2,...,{\alpha }_{i-1},{\alpha }_i^{\prime },{\alpha }_{i+1},...,{\alpha }_n)$. Then for any arbitrary $j\ne i$ ($1\le j\le n$), there exists ${\alpha }_j^{\prime }\in p_j\left(\mathfrak{M}\right)$ such that $\mathfrak{M}\vDash \neg H_0({\alpha }_1,{\alpha }_2,...,{\alpha }_{j-1},{\alpha }_j^{\prime },{\alpha }_{j+1},...,{\alpha }_n)$. Let us prove this.

Denote ${\overline{\alpha }}^{ij}:=\{{\alpha }_1,...,{\alpha }_n\}\setminus \{{\alpha }_i,{\alpha }_j\}$, $p_i^{\prime }:=tp({\alpha }_i/A{\overline{\alpha }}^{ij})$, $p_j^{\prime }:=tp({\alpha }_j/A{\overline{\alpha }}^{ij})$, and let $H_0^{ij}(x,y):=$$H_0({\alpha }_1,...{\alpha }_{j-1},y,{\alpha }_{j+1},...,{\alpha }_{i-1},x,{\alpha }_{i+1},...,{\alpha }_n)$. Since $\{p_1,p_2,...,p_n\}\setminus \left\{p_i\right\}$ and $\{p_1,p_2,...,p_n\}\setminus \left\{p_j\right\}$ are orthogonal families of 1-types, then $p_i\left(\mathfrak{M}\right)=p_i^{\prime }\left(\mathfrak{M}\right)$ and $p_j\left(\mathfrak{M}\right)=p_j^{\prime }\left(\mathfrak{M}\right)$. Since ${\alpha }_i\in H_0^{ij}(\mathfrak{M},{\alpha }_j)\cap p_i\left(\mathfrak{M}\right)$ and ${\alpha }_i^{\prime }\in \neg H_0^{ij}(\mathfrak{M},{\alpha }_j)\cap p_i\left(\mathfrak{M}\right)$, we have that $p_j^{\prime }{\cancel{\perp }}^w{p}_i^{\prime }$.

By Note 1 the relation of non-weak orthogonality is symmetric, therefore, $p_i^{\prime }{\cancel{\perp }}^w{p}_j^{\prime }$ and there exits ${\alpha }_j^{\prime }\in p_j^{\prime }\left(\mathfrak{M}\right)$ such that $\mathfrak{M}\vDash \neg H_0({\alpha }_1,{\alpha }_2,...,{\alpha }_{j-1},{\alpha }_j^{\prime },{\alpha }_{j+1},...,{\alpha }_n)$.

Denote ${\overline{\alpha }}_i:=({\alpha }_1,{\alpha }_2,...,{\alpha }_{i-1},{\alpha }_{i+1},...,{\alpha }_n)$. To simplify the notation, until the end of this proof, we move the free variable of H forward. Denote $H(x,{\overline{\alpha }}_i):=H({\alpha }_1,{\alpha }_{i-1},x,{\alpha }_{i+1},...,{\alpha }_n)$. Note that the free variable in the new notation can correspond to realizations of different $p_i$’s depending on i.

We construct an n-A-formula $H(x_1,x_2,...,x_n)$ such that $\mathfrak{M}\vDash H({\alpha }_1,{\alpha }_2,...,{\alpha }_n)$ and for each i ($1\le i\le n$) the following holds: $H(\mathfrak{M},{\overline{\alpha }}_i)\cap p_i\left(\mathfrak{M}\right)\ne \varnothing$, $\neg H(\mathfrak{M},{\overline{\alpha }}_i)\cap p_i\left(\mathfrak{M}\right)\ne \varnothing$, and either $\neg H(\mathfrak{M},{\overline{\alpha }}_i)=H{(\mathfrak{M},{\overline{\alpha }}_i)}^{+}$ or $\neg H(\mathfrak{M},{\overline{\alpha }}_i)=H{(\mathfrak{M},{\overline{\alpha }}_i)}^{-}$.

$(i+1)$ We define $H_{i+1}$ by induction on $i(0\le i\le n-1)$. Consider $H_i(\mathfrak{M},{\overline{\alpha }}_i)={\displaystyle \bigcup _k}{H}_{i,k}(\mathfrak{M},{\alpha }_i)$, where $H_{i,k}$ are all the convex parts of $H_i(x,{\overline{\alpha }}_i)$. Denote by $H_i^c(x,{\overline{\alpha }}_i)$ the convex part of $H_i(x,{\overline{\alpha }}_i)$ such that ${\alpha }_i\in H_{i,k}(\mathfrak{M},{\alpha }_i)$.

$H_{i+1}^{\prime }(y,{\overline{\alpha }}_i):=\forall x(H_i^c(x,{\overline{\alpha }}_i)\to x<y)$, if ${\alpha }_i<{\alpha }_i^{\prime }$. Here, $H_{i+1}^{\prime }(y,{\overline{\alpha }}_i)=H_i^c{(y,{\overline{\alpha }}_i)}^{+}$.

$H_{i+1}^{\prime }(y,{\overline{\alpha }}_i):=\forall x(H_i^c(x,{\overline{\alpha }}_i)\to y<x)$, if ${\alpha }_i^{\prime }<{\alpha }_i$. Here, $H_{i+1}^{\prime }(y,{\overline{\alpha }}_i)=H_i^c{(y,{\overline{\alpha }}_i)}^{-}$.

$H_{i+1}(y,{\overline{\alpha }}_i):=\forall x(H_{i+1}^{\prime }(x,{\overline{\alpha }}_i)\to y<x)$, if ${\alpha }_i<{\alpha }_i^{\prime }$. Here, $H_{i+1}^{\prime }(y,{\overline{\alpha }}_i)={\left(H_i^c{(y,{\overline{\alpha }}_i)}^{+}\right)}^{-}$.

$H_{i+1}(y,{\overline{\alpha }}_i):=\forall x(H_{i+1}^{\prime }(x,{\overline{\alpha }}_i)\to y<x)$, if ${\alpha }_i^{\prime }<{\alpha }_i$. Here, $H_{i+1}^{\prime }(y,{\overline{\alpha }}_i)={\left(H_i^c{(y,{\overline{\alpha }}_i)}^{-}\right)}^{+}$.

It follows from the definition of $H_{i+1}(y,{\overline{\alpha }}_i)$ that $\mathfrak{M}\vDash H_{i+1}({\alpha }_i,{\overline{\alpha }}_i)\wedge \neg H_{i+1}({\alpha }_i^{\prime },{\overline{\alpha }}_i)$.

Suppose that ${\alpha }_i<{\alpha }_i^{\prime }$. Denote $S^i\left({\overline{\alpha }}_i\right):=\forall y\left(H_{i+1}{(y,{\overline{\alpha }}_i)}^{+}\leftrightarrow \neg H_{i+1}(y,{\overline{\alpha }}_i)\right)$ and ${\overline{\alpha }}_i={\alpha }_j,{\overline{\alpha }}_{ij}$ for any $j\ne i(1\le j\le n)$. Since ${\overline{\alpha }}_i$ realizes a weakly orthogonal subset of $\Gamma$, then $\mathfrak{M}\vDash S^i({\alpha }_j,{\overline{\alpha }}_{ij})$ implies the existence of ${\mathsf{\Theta }}_j^i\left(x_j\right)\in p_j$ such that $\mathfrak{M}\vDash \forall x_j({\mathsf{\Theta }}_j^i\left(x_j\right)\to S^i(x_j,{\overline{\alpha }}_{ij}))$. For the case ${\alpha }_i^{\prime }<{\alpha }_i$, the formula $S^i\left({\overline{\alpha }}_i\right):=\forall y(H_{i+1}{(y,{\overline{\alpha }}_i)}^{-}\leftrightarrow \neg H_{i+1}(y,{\overline{\alpha }}_i))$ satisfies for convenient ${\mathsf{\Theta }}_j^i\left(x_j\right)\in p_j$, $\mathfrak{M}\vDash \forall x_j({\mathsf{\Theta }}_j^i\left(x_j\right)\to S^i(x_j,{\overline{\alpha }}_{ij}))$. Now define $H_{i+1}(x_j,{\overline{\alpha }}_j)$ for $j\ne i$:

if $j<i$, let $H_{i+1}(x_j,{\overline{\alpha }}_j):=H_{j+1}(x_j,{\overline{\alpha }}_j)$;

if $i<j$, let $H_{i+1}(x_j,{\overline{\alpha }}_j):=H_0(x_j,{\overline{\alpha }}_j)$. For each $j\ne i$ ($1\le j\le n$), let ${\mathsf{\Theta }}_j\left(x_j\right):={\displaystyle \underset{i}{\bigwedge }}{\mathsf{\Theta }}_j^i\left(x_j\right)$.

Denote $H(x_1,x_2,...,x_n):=H_n(x_1,x_2,...,x_n)\wedge {\displaystyle \underset{j}{\bigwedge }}{\mathsf{\Theta }}_j\left(x_j\right)$. Following the definition for each i ($1\le i\le n$), we have $\mathfrak{M}\vDash H({\alpha }_i,{\overline{\alpha }}_i)\wedge \neg H({\alpha }_i^{\prime },{\overline{\alpha }}_i)$ and either $\neg H(\mathfrak{M},{\overline{\alpha }}_i)=H{(\mathfrak{M},{\overline{\alpha }}_i)}^{+}$ or $\neg H(\mathfrak{M},{\overline{\alpha }}_i)=H{(\mathfrak{M},{\overline{\alpha }}_i)}^{-}$. □

Below, the notations from [11] are modified for consistency. For instance, the formula H in this article acts as the formula $\mathsf{\Phi }$ from [11]. And we let $H(\mathfrak{M},y)$ to be defining a convex set starting from negative infinity as a representation of the right border of $\mathsf{\Phi }(\mathfrak{M},y)$.

Let T be a weakly o-minimal theory, A be a finite and I be an infinite subset of an ${\left|I\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$. Let $H(x,y)$ be a convex A-formula such that for each $a\in I$, $H(\mathfrak{M},a)={\left(H{(\mathfrak{M},a)}^{+}\right)}^{-}$.

Definition 16

([10]). The formula $H(x,y)$ is tidy on I if one of the following holds:

(1) The formula H is locally increasing on I, that is, for each $a\in I$ there is an interval $J\subseteq I$ such that $a\in J$ and H strictly increases on J;

(2) The formula H is locally decreasing on I, that is, for each $a\in I$ there is an interval $J\subseteq I$ such that $a\in J$ and H strictly decreases on J;

(3) The formula H is locally constant on I, that is, for each $a\in I$ there is an interval $J\subseteq I$ such that $a\in J$ and H is constant on J.

Let $H(x,y)$ be a convex A-formula such that for each $a\in I$, $H(\mathfrak{M},a)={\left(H{(\mathfrak{M},a)}^{+}\right)}^{-}$. Denote

$${\psi }_0\left(y\right):=\exists y_1\exists y_2\left(y_1<y<y_2\wedge \forall z_1\forall z_2\left(y_1<z_1<z_2<y_2\to \forall x(H(x,z_1)\leftrightarrow H(x,z_2))\right)\right);$$

$${\psi }_1\left(y\right):=\exists y_1\exists y_2\left(y_1<y<y_2\wedge \forall z_1\forall z_2\left(y_1<z_1<z_2<y_2\to \exists t(\neg H(t,z_1)\wedge H(t,z_2))\right)\right);$$

$${\psi }_2\left(y\right):=\exists y_1\exists y_2\left(y_1<y<y_2\wedge \forall z_1\forall z_2\left(y_1<z_1<z_2<y_2\to \exists t(H(t,z_1)\wedge \neg H(t,z_2))\right)\right).$$

The formula ${\psi }_0\left(x\right)$ means that there is an interval around y on which H is constant. Analogically, ${\psi }_1$ and ${\psi }_2$ mean that on some interval around y the formula H increases or decreases, respectively. Thus, $\exists xH(x,\mathfrak{M})={\psi }_0\left(\mathfrak{M}\right)\cup {\psi }_1\left(\mathfrak{M}\right)\cup {\psi }_2\left(\mathfrak{M}\right)\cup X$ [10]. Since the theory T is weakly o-minimal, each ${\psi }_i\left(\mathfrak{M}\right)$ is a finite union of convex sets and X is a finite set. Thus, for any $i\in \{0,1,2\}$, ${\psi }_i\left(\mathfrak{M}\right)={\displaystyle \bigcup _j}{\psi }_i^j\left(\mathfrak{M}\right)$.

Below we study the behavior of the formula H on the convex formula ${\psi }_i^j$ and give the definition of the depth of H on ${\psi }_i^j$.

Let

$$\begin{array}{c}{\epsilon }_0^j(y_1,y_2):={\psi }_0^j\left(y_1\right)\wedge {\psi }_0^j\left(y_2\right)\wedge \forall z\left(\left(y_1\le z\le y_2\vee y_2\le z\le y_1\right)\to \forall x\left(H(x,y_1)\leftrightarrow H(x,z)\right)\right));\\ {\epsilon }_1^j(y_1,y_2):={\psi }_1^j\left(y_1\right)\wedge {\psi }_1^j\left(y_2\right)\wedge (y_1<y_2\to \forall z_1\forall z_2(y_1<z_1<z_2<y_2\to \\ \exists t_1\exists t_2\exists t_3(\neg H(t_1,y_1)\wedge H(t_1,z_1)\wedge \neg H(t_2,z_1)\wedge H(t_2,z_2)\wedge \neg H(t_3,z_2)\wedge H(t_3,y_2))\left)\right)\wedge (y_2<y_1\to \forall z_1\forall z_2(y_2<z_1<\\ z_2<y_1\to \exists t_1\exists t_2\exists t_3(\neg H(t_1,y_2)\wedge H(t_1,z_1)\wedge \neg H(t_2,z_1)\wedge H(t_2,z_2)\wedge \neg H(t_3,z_2)\wedge H(t_3,y_1))\left)\right);\end{array}$$

$$\begin{array}{c}{\epsilon }_2^j(y_1,y_2):={\psi }_2^j\left(y_1\right)\wedge {\psi }_2^j\left(y_2\right)\wedge (y_1<y_2\to \forall z_1\forall z_2(y_1<z_1<z_2<y_2\to \\ \exists t_1\exists t_2\exists t_3(\neg H(t_1,y_2)\wedge H(t_1,z_2)\wedge \neg H(t_2,z_2)\wedge H(t_2,z_1)\wedge \neg H(t_3,z_1)\wedge H(t_3,y_1))\left)\right)\wedge (y_2<y_1\to \forall z_1\forall z_2(y_2<z_1<\\ z_2<y_1\to \exists t_1\exists t_2\exists t_3(\neg H(t_1,y_1)\wedge H(t_1,z_2)\wedge \neg H(t_2,z_2)\wedge H(t_2,z_1)\wedge \neg H(t_3,z_1)\wedge H(t_3,y_2))\left)\right).\end{array}$$

These three relations indicate pairs of elements, on the intervals between which the formula H is constant (${\epsilon }_0^j$), strictly increases (${\epsilon }_1^j$), or strictly decreases (${\epsilon }_2^j$). From the definitions it follows that for each $i\in \{0,1,2\}$, $\mathfrak{M}\vDash \forall y\left({\psi }_i\left(y\right)\to \exists y^{\prime }{\epsilon }_i^j(y,y^{\prime })\right)$. For each $a\in M$, the sets ${\epsilon }_0^j(\mathfrak{M},a)$, ${\epsilon }_1^j(\mathfrak{M},a)$, ${\epsilon }_2^j(\mathfrak{M},a)$ are maximal convex sets containing a, on which H is constant, strictly increases or strictly decreases, respectively. Next, we look at ${\epsilon }_i^j(\mathfrak{M},a)$, where $i\in \{0,1,2\}$, and a belongs to the convex part ${\psi }_i^j\left(\mathfrak{M}\right)$ of ${\psi }_i\left(\mathfrak{M}\right)$.

We consider the following three cases.

$i=0$. Let ${\psi }_0^j\left(y\right)$ be an arbitrary convex part of the formula ${\psi }_0\left(y\right)$. If for some $\alpha \in {\psi }_0^j\left(\mathfrak{M}\right)$, we have ${\psi }_0^j\left(\mathfrak{M}\right)={\epsilon }_0^j(\mathfrak{M},\alpha )$, then in this case we say, that the depth of the formula $H(x,y)$ on ${\psi }_0^j\left(\mathfrak{M}\right)$ is equal to zero. If the set of all ${\epsilon }_0^j$-classes is finite, then because each ${\epsilon }_0^j$-class is A-definable, the depth of the A-formula $H(x,y)$ on each ${\epsilon }_0^j$-class is equal to zero. If the set of all ${\epsilon }_0^j-$classes is infinite, then the linear ordering on these classes is dense. In the paper, we assume that the order is dense since discrete ordering implies maximality of the number of countable non-isomorphic models [15,16]. In this case, we say that the depth of the formula H on ${\psi }_0^j$ is at least 1. Notice that then the formula ${\psi }_0^j\left(x\right)$ can be refined to A-formulas ${\psi }_{01}^j\left(x\right)$ and ${\psi }_{02}^j\left(x\right)$ in the following way.

We say that $\mathfrak{M}\vDash {\psi }_{01}^j\left(\alpha \right)$ if $\mathfrak{M}\vDash {\psi }_0^j\left(\alpha \right)$ and there is an interval around $\alpha$ on which H increases up to ${\epsilon }_0$. And let ${\epsilon }_{01}^j(\mathfrak{M},\alpha )$ be the maximal convex set containing $\alpha$ on which H increases up to ${\epsilon }_0^j$. That is, if $\beta \in {\epsilon }_{01}^j(\mathfrak{M},\alpha )$ and $\alpha <\beta$, then $\mathfrak{M}\vDash {\psi }_0^j\left(\beta \right)$ and $H(\mathfrak{M},\alpha )⊊H(\mathfrak{M},\gamma )⊊H(\mathfrak{M},\beta )$ for each $\gamma$ with ${\epsilon }_0^j(\mathfrak{M},\alpha )<{\epsilon }_0^j(\mathfrak{M},\gamma )<{\epsilon }_0^j(\mathfrak{M},\beta )$.

We say that $\mathfrak{M}\vDash {\psi }_{02}^j\left(\alpha \right)$ if $\mathfrak{M}\vDash {\psi }_0^j\left(\alpha \right)$ and there is an interval around $\alpha$ on which H decreases up to ${\epsilon }_0^j$. And let ${\epsilon }_{02}(\mathfrak{M},\alpha )$ be the maximal convex set containing $\alpha$ on which H decreases up to ${\epsilon }_0^j$. That is, if $\beta \in {\epsilon }_{02}^j(\mathfrak{M},\alpha )$ and $\alpha <\beta$, then $\mathfrak{M}\vDash {\psi }_0^j\left(\beta \right)$ and $H(\mathfrak{M},\beta )⊊H(\mathfrak{M},\gamma )⊊H(\mathfrak{M},\alpha )$ for each $\gamma$ with ${\epsilon }_0^j(\mathfrak{M},\alpha )<{\epsilon }_0^j(\mathfrak{M},\gamma )<{\epsilon }_0^j(\mathfrak{M},\beta )$.

$i=1$. Let ${\psi }_1^j\left(y\right)$ be a convex part of the formula ${\psi }_1\left(y\right)$. If for some $\alpha \in {\psi }_1^j\left(\mathfrak{M}\right)$, ${\epsilon }_1^j(\mathfrak{M},\alpha )={\psi }_1^j\left(\mathfrak{M}\right)$, we say that the depth of the formula H on ${\psi }_1^j$ is equal to 1.

If ${\alpha }_1,{\alpha }_2,...,{\alpha }_m\in {\psi }_1^j\left(\mathfrak{M}\right)$, $i<\omega$, are representatives of distinct ${\epsilon }_1^j$-classes such that ${\displaystyle \bigcup _{1\le k\le m}}{\epsilon }_1^j(\mathfrak{M},{\alpha }_k)={\psi }_1^j\left(\mathfrak{M}\right)$, then denote ${\psi }_1^{j,k}\left(y\right):={\epsilon }_1^j(y,{\alpha }_k)$. Since there is only a finite number of ${\epsilon }_1^j$-classes, each of them is A-definable. Then for each k, $1\le k\le m$ the depth of H on ${\psi }_1^{j,k}\left(y\right)$ is equal to 1.

If the set of ${\epsilon }_1^j$-classes is infinite, then the linear ordering on these classes is dense and the depth of the formula ${\psi }_1\left(x\right)$ is greater than or equal to 2. Consider two elements $\alpha ,\beta \in {\psi }_1^j\left(\mathfrak{M}\right)$ such that $\alpha <\beta$ and $\mathfrak{M}\vDash \neg {\epsilon }_1^j(\alpha ,\beta )$. Suppose $H(\mathfrak{M},\alpha )\subset H(\mathfrak{M},\beta )$. If for each $\gamma$, $\alpha <\gamma <\beta$ implies $H(\mathfrak{M},\alpha )\subset H(\mathfrak{M},\gamma )\subset H(\mathfrak{M},\beta )$, then $\beta \in {\epsilon }_1^j(\mathfrak{M},\alpha )$, what contradicts the initial condition. So, there is ${\gamma }_1$ such that $\alpha <{\gamma }_1<\beta$ and $H(\mathfrak{M},{\gamma }_1)\subset H(\mathfrak{M},\alpha )$. Assume that for each ${\gamma }_i$ there exist ${\beta }_{i-1},{\beta }_i$ such that $\alpha <{\beta }_i<{\gamma }_i<{\beta }_{i-1}<\beta$, $\mathfrak{M}\vDash \neg {\epsilon }_1^j(\alpha ,{\beta }_i)$. and $H(\mathfrak{M},\alpha )\subset H(\mathfrak{M},{\beta }_i)$. Then the formula $\alpha <y\wedge H(\mathfrak{M},y)\supset H(\mathfrak{M},\alpha )$ has an infinite number of elements $\left\{{\beta }_i\right|i<\omega \}$ which satisfy it and alternate with the elements from the set $\left\{{\gamma }_i\right|i<\omega \}$. This contradicts the definition of a weakly o-minimal theory. Thus

$$\begin{array}{c}\hfill \mathrm{for}\mathrm{each}\alpha \in {\psi }_1^j\left(\mathfrak{M}\right)\mathrm{there}\mathrm{is}\gamma \in {\psi }_1^j\left(\mathfrak{M}\right)\mathrm{such}\mathrm{that}\\ \hfill \mathfrak{M}\vDash \alpha <\gamma \wedge \neg {\epsilon }_1^j(\alpha ,\gamma )\wedge H(\mathfrak{M},\gamma )\subset H(\mathfrak{M},\alpha )\wedge \\ \hfill \forall y\left((\alpha <y<\gamma \wedge \neg {\epsilon }_1^j(\alpha ,y))\to H(\mathfrak{M},y)\subset H(\mathfrak{M},\alpha )\right).\end{array}$$

(1)

Introduce a new formula $H^1(x,y):=\exists z({\epsilon }_1^j(z,y)\wedge H(x,z))$. This formula is locally constant on ${\psi }_1^j\left(\mathfrak{M}\right)$. From (1) it follows that $H^1$ is decreasing on ${\psi }_1^j/{\epsilon }_1^j$. Consider the formula ${\epsilon }_{12}^j(x,y):=(x<y\wedge \neg {\epsilon }_1^j(x,y)\wedge \forall z((x<z<y\wedge \neg {\epsilon }_1^j(x,z)\wedge \neg {\epsilon }_1^j(z,y))\to H(\mathfrak{M},x)\supset H(\mathfrak{M},z)\supset H(\mathfrak{M},y)))\wedge (y<x\wedge \neg {\epsilon }_1^j(x,y)\wedge \forall z((y<z<x\wedge \neg {\epsilon }_1^j(x,z)\wedge \neg {\epsilon }_1^j(z,y))\to H(\mathfrak{M},y)\supset H(\mathfrak{M},z)\supset H(\mathfrak{M},x)))$. This formula is a relation of equivalence ${\psi }_1^j\left(\mathfrak{M}\right)$ and each ${\epsilon }_{12}^j$-class contains an infinite number of ${\epsilon }_1^j$-classes. If ${\epsilon }_{12}^j(\mathfrak{M},\alpha )={\psi }_1^j\left(\mathfrak{M}\right)$ for some $\alpha \in {\psi }_1^j\left(\mathfrak{M}\right)$, then we say that the depth of H on ${\psi }_1^j\left(\mathfrak{M}\right)$ is equal to 2.

If there exists a finite number of elements ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m\in {\psi }_1^j\left(\mathfrak{M}\right)$ from different ${\epsilon }_{12}^j$-classes such that ${\bigcup }_{1\le k\le m}{\epsilon }_{12}^j(\mathfrak{M},{\alpha }_k)={\psi }_1^j\left(\mathfrak{M}\right)$, then denote ${\psi }_1^{j,k}\left(y\right):={\epsilon }_{12}^j(y,{\alpha }_k)$ for each $1\le k\le m$. Since the number of ${\epsilon }_{12}^j$-classes is finite, each of these classes is definable. Then for each k, $1\le k\le m$ the depth of H on ${\psi }_1^{j,k}\left(y\right)$ is equal to 2.

If the set of ${\epsilon }_{12}^j-$classes is infinite, then the linear ordering on these classes is dense and the depth of the formula ${\psi }_1^j\left(x\right)$ is greater than or equal to 3.

$i=2$. This case is analogical to the case $i=1$, instead of decreasing increasing classes we have increasing decreasing classes from the same consideration based on the next condition (2):

$$\begin{array}{c}\hfill \mathrm{for}\mathrm{each}\alpha \in {\psi }_2^j\left(\mathfrak{M}\right)\mathrm{there}\mathrm{is}\gamma \in {\psi }_2^j\left(\mathfrak{M}\right)\mathrm{such}\mathrm{that}\\ \hfill \mathfrak{M}\vDash \alpha <\gamma \wedge \neg {\epsilon }_2^j(\alpha ,\gamma )\wedge H(\mathfrak{M},\alpha \subset H(\mathfrak{M},\gamma )\wedge \\ \hfill \forall y\left((\alpha <y<\gamma \wedge \neg {\epsilon }_2^j(\alpha ,y))\to H(\mathfrak{M},\alpha )\subset H(\mathfrak{M},y)\right).\end{array}$$

(2)

Each ${\epsilon }_2^j$-class consists of ${\epsilon }_{21}^j$-classes such that the formula H decreases on ${\epsilon }_2^j(\mathfrak{M},\alpha )$ and increases on representatives of distinct ${\epsilon }_{21}^j$-classes of elements from ${\epsilon }_2^j(\mathfrak{M},\alpha )$.

Let $\alpha \in {\psi }_i^j\left(\mathfrak{M}\right)$. Consider the three next formulas:

${\psi }_1^j\left(y\right)\wedge \alpha <y\wedge \neg {\epsilon }_1^j(\alpha ,y)\wedge H(\mathfrak{M},y)\supset H(\mathfrak{M},\alpha )$;

${\psi }_2^j\left(y\right)\wedge \alpha <y\wedge \neg {\epsilon }_2^j(\alpha ,y)\wedge H(\mathfrak{M},\alpha )\supset H(\mathfrak{M},y)$;

${\psi }_0^j\left(y\right)\wedge \alpha <y\wedge \neg {\epsilon }_0^j(\alpha ,y)\wedge H(\mathfrak{M},y)\supset H(\mathfrak{M},\alpha )$.

Since the theory is weakly o-minimal, in the case ${\psi }_i^j\left(\mathfrak{M}\right)\ne \varnothing$, each of these formulas has a finite number of convex parts. This number determines the depth of the formula H on ${\psi }_i^j\left(\mathfrak{M}\right)$ [11]. Thus, there is only a finite number of equivalence relations $\left\{E_k(x,y)\right|1\le k\le n\}$ with convex classes on ${\psi }_i^j\left(\mathfrak{M}\right)$ such that for each k ($1\le k\le n$), each $E_k$-class contains an infinite number of $E_{k-1}$-classes, the number of $E_n$-classes is finite, and for each $k\ge 1$ $E_k$ is some ${\epsilon }_{i_1{i}_2\dots i_k}^j$, where $i_t\in \{1,2\}$ and $i_t=1$ ($i_t=2$) implies $i_{t+1}=2$ ($i_{t+1}=1$) for all $1\le t\le k$.

Then there exists a finite number of elements ${\alpha }_1,{\alpha }_2,...,{\alpha }_m\in {\psi }_1^j\left(\mathfrak{M}\right)$ from different $E_n$-classes such that ${\displaystyle \bigcup _{1\le s\le m}}{E}_n(\mathfrak{M},{\alpha }_s)={\psi }_1^j\left(\mathfrak{M}\right)$. Thus, denote ${\psi }_1^{j,s}\left(y\right):=E_n(y,{\alpha }_s)$. Since there is a finite number of $E_n$-classes, each of these classes is definable. Then for each s ($1\le s\le m$) the depth of H on ${\psi }_1^{j,s}\left(y\right)$ is equal to n.

The formula $H^n(x,y):=\exists z(E_{n-1}(z,y)\wedge H(x,z))$ is strictly monotonic on ${\psi }_i^{j,s}/E_{n-1}$. This means if $\alpha ,\beta \in {\psi }_i^{j,s}\left(\mathfrak{M}\right)$ are such that $\mathfrak{M}\vDash \neg E_{n-1}(\alpha ,\beta )\wedge \alpha <\beta$, then $\mathfrak{M}\vDash H^n(\mathfrak{M},\beta )\supset H^n(\mathfrak{M},\alpha )$ implies $\mathfrak{M}\vDash \forall x\forall y(({\psi }_i^{j,s}\left(x\right)\wedge {\psi }_i^{j,s}\left(y\right)\wedge \neg E_{n-1}(x,y)\wedge x<y)\to H^n(\mathfrak{M},y)\supset H^n(\mathfrak{M},x))$, and $\mathfrak{M}\vDash H^n(\mathfrak{M},\alpha )\supset H^n(\mathfrak{M},\beta )$ implies $\mathfrak{M}\vDash \forall x\forall y(({\psi }_i^{j,s}\left(x\right)\wedge {\psi }_i^{j,s}\left(y\right)\wedge \neg E_{n-1}(x,y)\wedge x<y)\to H^n(\mathfrak{M},x)\supset H^n(\mathfrak{M},y))$.

Thus, taking into consideration the definition of the depth of the 2-formula H, we obtain the following.

Proposition 2.

Let $p,q\in S_1\left(A\right)$ be two non-isolated 1-types over A, and H be a 2-A-formula such that for each $\alpha \in p\left(\mathfrak{M}\right)$, we have ${\left(H{(\mathfrak{M},\alpha )}^{+}\right)}^{-}=H(\mathfrak{M},\alpha )$, $H(\mathfrak{M},\alpha )\cap q\left(\mathfrak{M}\right)\ne \varnothing$, and $\neg H(\mathfrak{M},\alpha )\cap q\left(\mathfrak{M}\right)\ne \varnothing$. Then there exists k ($1\le k\le n$), where n is the depth of H on the corresponding ${\psi }_i^j\left(y\right)$, such that $H^k$ is strictly monotonic on $\mathsf{\Theta }/E_{k-1}$ for some convex formula $\mathsf{\Theta }\left(y\right)\in p$, which is a subformula of some $E_k$-class, and for each $\alpha \in p\left(\mathfrak{M}\right)$, $E_{k-1}(\mathfrak{M},\alpha )\subset p\left(\mathfrak{M}\right)$.

In Ref. [11], a natural generalization of the notions from [10] was given.

Definition 17

([11]). If H and I are as in Definition 16, then H is n-tidy on I if the following holds:

(1) For each $a,b,c\in I$ if $a<b<c$ and $H(\mathfrak{M},a)=H(\mathfrak{M},c)$, then $H(\mathfrak{M},a)=H(\mathfrak{M},b)$;

(2) The formula $H^n$ is tidy on $I/E_{n-1}$, where $H^n(x,y):=\exists z(E_{n-1}(y,z)\wedge H(x,z))$;

(3) For each $a\in I$ $E_n(I,a)/E_{n-1}$ has no minimum and no maximum;

(4) $|I/E_n| \ge {\aleph }_0$;

Where $E_n$ is an equivalence relation on I such that for each $a,b\in I$ $\mathfrak{M}\vDash E_n(b,a)$ if and only if either $\mathfrak{M}\vDash E_{n-1}(b,a)$, or $\mathfrak{M}\vDash \neg E_{n-1}(a,b)\wedge a<b$ implies that $(H^n\upharpoonright [a,b])/E_{n-1}$ is strictly monotonic, or $\mathfrak{M}\vDash \neg E_{n-1}(a,b)\wedge b<a$ implies that $(H^n\upharpoonright [b,a])/E_{n-1}$ is strictly monotonic. And 0-tidy is tidy, $H^0(x,y):=H(x,y)$, and $E_0(x,y):=x=y$.

Definition 18

([11]). If H and I are as in Definition 16, the formula H is strongly tidy on I if there exists $n<\omega$ such that H is $(n-1)$-tidy on I and $H^n$ is strictly monotonic on $I/E_{n-1}$. Then the depth of H on I equals n.

The notion of the depth of a formula was applied in studying o-stable ordered groups of finite convexity rank [20].

Definition 19.

A weakly o-minimal structure $\mathfrak{M}$ is said to have strong monotonicity [11] (monotonicity [10]) if for each convex formula $H(x,y,\overline{a})$ with $\mathfrak{M}\vDash \forall xH(\mathfrak{M},x,\overline{a})={\left(H{(\mathfrak{M},x,\overline{a})}^{+}\right)}^{-}$, there is $m<\omega$ and a partition of $dom\left(H(\mathfrak{M},y,\overline{a})\right)$ into definable sets $X,I_1,\dots ,I_m$ such that X is finite, each $I_i$ is convex, and the formula $H(x,y,\overline{a})$ is strongly tidy (tidy) on each $I_i$.

Definition 20

([11]). Let $\mathfrak{M}$ be a weakly o-minimal structure. Then $\mathfrak{M}$ has finite depth, if for each convex formula $H(x,y,\overline{z})$ with $\mathfrak{M}\vDash \forall x\forall \overline{z}H(\mathfrak{M},x,\overline{z})={\left(H{(\mathfrak{M},x,\overline{z})}^{+}\right)}^{-}$ there is $n<\omega$ such that for each $\overline{a}\in M$ and each convex set $I\subseteq dom\left(H(\mathfrak{M},y,\overline{a})\right)$ on which $H(x,y,\overline{a})$ is strongly tidy, the depth of $H(x,y,\overline{a})$ on I is less than n.

Theorem 6

([10]). If all models of $Th\left(\mathfrak{M}\right)$ are weakly o-minimal, then $\mathfrak{M}$ has monotonicity.

Theorem 7

([11]). If all models of $Th\left(\mathfrak{M}\right)$ are weakly o-minimal, then $\mathfrak{M}$ has strong monotonicity and finite depth.

Proposition 3.

Weakly o-minimal theories are theories with monotonic non-orthogonality on equivalence classes.

Proof of Proposition 3.

Let $A\subseteq \mathfrak{M}$ be finite, $\mathfrak{M}\vDash T$ be ${\left|A\right|}^{+}$-saturated, $n<\omega$, and $\Gamma =\{p_1,p_2,\dots ,p_n\}$ be a non-weakly orthogonal family of non-isolated types from $S_1\left(A\right)$ such that each ${\Gamma }^{\prime }⊊\Gamma$ is weakly orthogonal. Since T is weakly o-minimal, each of the types from $\Gamma$ has a convex set of all realizations.

Since $\Gamma$ is non-weakly orthogonal, take the formula $H(y_1,y_2,\dots ,y_n)$ from Proposition 1. Then for all $1\le i\le n$, the formula $H(y_1,y_2,\dots ,y_n)$ is convex in $y_i$ and for all ${\alpha }_1\in p_1^c\left(\mathfrak{M}\right),{\alpha }_2\in p_2\left(\mathfrak{M}\right),\dots ,{\alpha }_{i-1}\in p_{i-1}\left(\mathfrak{M}\right),p_{i+1}\left(\mathfrak{M}\right),\dots ,{\alpha }_n\in p_n\left(\mathfrak{M}\right)$,

$$H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)\cap p_i\left(\mathfrak{M}\right)\ne \varnothing ,$$

$$\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)\cap p_i\left(\mathfrak{M}\right)\ne \varnothing , \mathrm{and} \mathrm{either}$$

$$\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)=H{({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)}^{+} \mathrm{or}$$

$$\neg H({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)=H{({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{i-1},\mathfrak{M},{\alpha }_{i+1},\dots ,{\alpha }_n)}^{-}.$$

By Proposition 2, for the formula $H_{1,i}^{\overline{\alpha }}(y_i,y_1)$, there exists an $A\overline{\alpha }$-formula $\mathsf{\Theta }\left(y_1\right)\in tp({\alpha }_1/A\overline{\alpha })=p_1^{\prime }$ and an $A\cup \overline{\alpha }$-definable equivalence relation $E_1$ such that $H_{1,i}^{\overline{\alpha }}(y_i,y_1)/E_1$ is strictly monotonic on $\mathsf{\Theta }\left(\mathfrak{M}\right)$. Notice that since $tp(\overline{\alpha }/A){\perp }^w{p}_1$, we have that $p_1\left(\mathfrak{M}\right)=p_1^{\prime }\left(\mathfrak{M}\right)$. Then $H_{1,i}^{{\overline{\alpha }}^{\prime }}(y_i,y_1)$ is strictly monotonic on $\mathsf{\Theta }\left(\mathfrak{M}\right)/E_1(x,y,{\overline{\alpha }}^{\prime })$ for each ${\overline{\alpha }}^{\prime }=({\alpha }_2^{\prime },\dots ,{\alpha }_{i-1}^{\prime },{\alpha }_{i+1}^{\prime },\dots ,{\alpha }_n^{\prime })\in \left(p_2\left(\mathfrak{M}\right)\times \dots \times p_{i-1}\left(\mathfrak{M}\right)\times p_{i+1}\left(\mathfrak{M}\right)\dots \times p_n\left(\mathfrak{M}\right)\right)$.

The same holds when an arbitrary pair $j\ne k$, are taken instead of 1 and i. □

Because of Proposition 3 all the results in the previous sections regarding theories with monotonic non-orthogonality on equivalence classes hold for weakly o-minimal theories. In particular, Proposition 3 implies the following.

Corollary 5.

Let T be a weakly o-minimal theory, and let Γ be an infinite family of pairwise weakly orthogonal non-isolated 1-types over a finite set A. Then $T(T,{\aleph }_0)=2^{{\aleph }_0}$.

Corollary 6.

Let T be a weakly o-minimal theory having less than $2^{{\aleph }_0}$ models. Then there is no infinite family of pairwise weakly orthogonal 1-types over a finite set.

Definition 21.

([12]) Let $p\in S_1\left(A\right)$, $B\subset M$ such that $\mathfrak{M}$ is ${|B\cup A|}^{+}$-saturated. Then a neighborhood of the set B in the type p is the following set:

$V_p\left(B\right):=\{\gamma \in M\mid$ there exist ${\gamma }_1,{\gamma }_2\in p\left(\mathfrak{M}\right)$ and a formula $H(x,\overline{b},\overline{c}),\overline{b}\in B,\overline{c}\in A$, such that ${\gamma }_1<H(\mathfrak{M},\overline{b},\overline{c})<{\gamma }_2,\gamma \in H(\mathfrak{M},\overline{b},\overline{c})\}$.

Let $\overline{\alpha }=\langle {\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k\rangle$ then $V_p\left(\overline{\alpha }\right):=V_p\left(\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k\}\right)$.

Note that for each $\alpha \in p\left(\mathfrak{M}\right)$, $V_p\left(\alpha \right)={\displaystyle \bigcup _{\phi \in CRF\left(p\right)\cup CLF\left(p\right)}}\phi (\mathfrak{M},\alpha )$. Indeed, if $H(x,\alpha )$ is an $(A\cup \{\alpha \left\}\right)$-definable formula as in the definition, then, if non-empty, ${\left(H{(x,\alpha )}^{+}\right)}^{-}\wedge (x\ge \alpha )$ is a p-preserving convex to the right formula, and ${\left(H{(x,\alpha )}^{-}\right)}^{+}\wedge (x\le \alpha )$ is a p-preserving convex to the left formula.

Note 2

([12]). Let $p,q\in S_1\left(A\right)$. Then p is almost orthogonal to q (denoted $p{\perp }^{a}q$) if and only if for some (equivalently, for every) $\alpha \in p\left(\mathfrak{M}\right)$, $V_q\left(\alpha \right)=\varnothing$.

The following theorem was previously proven in [12]. Below, we present an alternative proof of this theorem, along with its important corollary.

Theorem 8.

Let T be a weakly o-minimal theory, let $p,q\in S_1\left(A\right)$ for some subset A of an ${\left|A\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$. Then if $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$ and p is social, then $p \cancel{\perp }{}_{ }{}^{a }q_{ }^{ }$ and q is social.

Proof of Theorem 8.

Let $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$ and p be social. Then there is a convex A-formula $H(x,y)$ such that for each $\alpha \in p\left(\mathfrak{M}\right)$ the sets $H(\mathfrak{M},\alpha )\cap q\left(\mathfrak{M}\right)$ and $\neg H(\mathfrak{M},\alpha )\cap q\left(\mathfrak{M}\right)$ are non-empty, $\neg H(\mathfrak{M},\alpha )=H{(\mathfrak{M},\alpha )}^{+}$, or, equivalently, $H(\mathfrak{M},\alpha )={(\neg H(\mathfrak{M},\alpha ))}^{-}$.

For each $\phi \in \left(CRF\right(p)\cup CLF(p\left)\right)$, denote $S_{\phi }(x,\alpha ):=\exists y(\phi (y,\alpha )\wedge H(x,y))$. Fix arbitrary $\alpha \in p\left(\mathfrak{M}\right)$.

(a) Suppose that for some $\phi \in CRF\left(p\right)$ either $S_{\phi }(\mathfrak{M},\alpha )\setminus H(\mathfrak{M},\alpha )\ne \varnothing$ or $H(\mathfrak{M},\alpha )\setminus S_{\phi }(\mathfrak{M},\alpha )\ne \varnothing$, without loss of generality suppose the first. Then the formula $S_{\phi }(x,\alpha )\wedge \neg H(x,\alpha )$ defines a bounded subset of $q\left(\mathfrak{M}\right)$. Since $\alpha$ is an arbitrary realization of p, then $p \cancel{\perp }{}_{ }{}^{a }q_{ }^{ }$.

(b) Now assume that for each ${\phi }_1,{\phi }_2\in (CRF\left(p\right)\cup CLF\left(p\right))$, $S_{{\phi }_2}(\mathfrak{M},\alpha )=S_{{\phi }_1}(\mathfrak{M},\alpha )=H(\mathfrak{M},\alpha )$. Denote $E(y_1,y_2):=\forall x(H(x,y_1)\leftrightarrow H(x,y_2))$. Since the theory T is weakly o-minimal, $E(\mathfrak{M},\alpha )={\displaystyle \bigcup _{1\le i\le n}}{E}_i(\mathfrak{M},\alpha )$ for some $n<\omega$ and some $(A\cup \{\alpha \left\}\right)$-definable sets $E_i(\mathfrak{M},\alpha )$. Let $I:=\{i\le n|E_i(\mathfrak{M},\alpha )\cap V_p\left(\alpha \right)\ne \varnothing \}$. Denote $E^{\prime }(x,\alpha ):={\displaystyle \underset{i\in I}{\bigvee }}{E}_i(x,\alpha )$. Then, by the assumption in the beginning of this paragraph, $V_p\left(\alpha \right)\subseteq {\left(E^{\prime }(\mathfrak{M},\alpha )\right)}^c$. By monotonicity, ${\gamma }_1<{\left(E^{\prime }(\mathfrak{M},\alpha )\right)}^c<{\gamma }_2$ for some realizations ${\gamma }_1,{\gamma }_2$ of p. Then we obtain a contradiction since no such definable set can be greater than $V_p\left(\alpha \right)$.

Next, we prove that the type q is social. Without loss of generality let H guarantee that $p \cancel{\perp }{}_{ }{}^{a }q_{ }^{ }$. By Note 1 $q \cancel{\perp }{}_{ }{}^{a }p_{ }^{ }$. Let $H_0$ be a convex formula guaranteeing $q \cancel{\perp }{}_{ }{}^{a }p_{ }^{ }$. Fix realizations $\alpha$ and $\beta$ of these types such that $\beta \in H(\mathfrak{M},\alpha )$ and $\alpha \in H_0(\beta ,\mathfrak{M})$. Towards a contradiction suppose that the type q is not social. Let $E_q(x,y)$ be an A-definable equivalence relation generated by the greatest convex to the right formula on q. Then $E_q(\mathfrak{M},\beta )$ is a convex proper subset of $q\left(\mathfrak{M}\right)$ bounded above and below by some realizations of q. Next in the proof, starting from $E_q(\mathfrak{M},\beta )$, we will successively apply certain formulas forming new definable sets over $(A\cup \{\beta \left\}\right)$ to obtain a contradiction.

Denote $H_0^{\prime }(\beta ,y):=\exists x\left(E_q(x,\beta )\wedge H_0(x,y)\right)$. Then $H_0^{\prime }(\beta ,\mathfrak{M})={\displaystyle \bigcup _{{\beta }^{\prime }\in E_q(\mathfrak{M},\beta )}}{H}_0({\beta }^{\prime },\mathfrak{M})⊊p\left(\mathfrak{M}\right)$ is a bounded subset of $p\left(\mathfrak{M}\right)$. Note that $\alpha \in H_0(\beta ,\mathfrak{M})\subseteq H_0^{\prime }(\beta ,\mathfrak{M})$.

Let $\epsilon$ be the equivalence relation on $p\left(\mathfrak{M}\right)$ generated by some convex to the right on p formula $\phi$. Denote ${\epsilon }^{\prime }(x,\beta ):=\exists y\left(H_0^{\prime }(\beta ,y)\wedge \epsilon (x,y)\right)$. Then ${\epsilon }^{\prime }(\mathfrak{M},\beta )={\displaystyle \bigcup _{{\alpha }^{\prime }\in H_0^{\prime }(\beta ,\mathfrak{M})}}\epsilon (\mathfrak{M},{\alpha }^{\prime })⊊p\left(\mathfrak{M}\right)$ is a bounded subset of $p\left(\mathfrak{M}\right)$. Note that, since the type p is social, there is an infinite number of distinct convex to the right formulas on p. If ${\phi }_1,{\phi }_2\in CRF\left(p\right)$ are such that ${\phi }_2$ is strictly greater than ${\phi }_1$, and ${\epsilon }_{{\phi }_1}$${\epsilon }_{{\phi }_2}$ be the equivalence relations on p generated by ${\phi }_1$ and ${\phi }_2$, then ${\epsilon }_{{\phi }_1}(\mathfrak{M},\alpha )⊊{\epsilon }_{{\phi }_2}(\mathfrak{M},\alpha )$, and therefore ${\epsilon }_{{\phi }_1}^{\prime }(\mathfrak{M},\beta )\subseteq {\epsilon }_{{\phi }_2}^{\prime }(\mathfrak{M},\beta )$. Because $\alpha \in H_0^{\prime }(\beta ,\mathfrak{M})$, for each $\gamma \in V_p\left(\alpha \right)$ there is $\psi \in CRF\left(p\right)$ such that $\gamma \in {\epsilon }_{\psi }^{\prime }(\mathfrak{M},\beta )$. The same also holds for each ${\psi }^{\prime }\in CRF\left(p\right)$ greater than $\psi$. We have

$$V_p\left(\alpha \right)\subseteq \bigcup _{\phi \in CRF\left(p\right)}{\epsilon }_{\phi }^{\prime }(\mathfrak{M},\beta ).$$

(3)

For each equivalence relation $\epsilon$ generated by a convex to the right formula on p denote $H^{\epsilon }(\beta ,y):=\exists x\left({\epsilon }^{\prime }(x,\beta )\wedge H(x,y)\right)$. Then $H^{\epsilon }(\beta ,\mathfrak{M})={\displaystyle \bigcup _{{\alpha }^{\prime }\in {\epsilon }^{\prime }(\mathfrak{M},\beta )}}H({\alpha }^{\prime },\mathfrak{M})\subseteq q\left(\mathfrak{M}\right)$ is a bounded subset of $q\left(\mathfrak{M}\right)$. Moreover, $E_q(\mathfrak{M},\beta )\subseteq H^{\epsilon }(\beta ,\mathfrak{M})$. But since $E_q$ is the maximal convex to the right equivalence relation on q, $E_q(\mathfrak{M},\beta )=H^{\epsilon }(\beta ,\mathfrak{M})$ for each $\epsilon$. Then $E_q(\mathfrak{M},\beta )={\displaystyle \bigcup _{\phi \in CRF\left(p\right)}}{H}^{{\epsilon }_{\phi }}(\beta ,\mathfrak{M})$.

Denote $S(y,\alpha ):=\exists x\left(H(x,\alpha )\wedge \forall x^{\prime }\left(H(x^{\prime },y)\to E_q(x^{\prime },x)\right)\right)$. Then

$$V_p\left(\alpha \right)\subseteq \bigcup _{\phi \in CRF\left(p\right)}{\epsilon }_{\phi }^{\prime }(\mathfrak{M},\beta )\subseteq S(\mathfrak{M},\alpha ).$$

From the definition it follows that $V_p\left(\alpha \right)$ cannot be a proper subset of any $(A\cup \{\alpha \left\}\right)$-definable bounded in $p\left(\mathfrak{M}\right)$ set. Nor can it be equal to any such set, since p is social and $V_p\left(\alpha \right)$ is not $(A\cup \{\alpha \left\}\right)$-definable. Thus, we have obtained a contradiction and the theorem is proved. □

Corollary 7.

Let T be a weakly o-minimal theory, let $p,q\in S_1\left(A\right)$ for some subset A of an ${\left|A\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$. Then if $p{\perp }^{a}q$ and $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$, then the type p is quasisolitary and the depth of the convex formula guaranteeing $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$ on some formula from $p\left(\mathfrak{M}\right)$ is equal to 1.

Proof of Corollary 7.

First part of the corollary follows directly from Theorem 8, 2.

Let $p,q\in S_1\left(A\right)$ be two non-isolated 1-types such that $p{\perp }^{a}q$, $p \cancel{\perp }{}_{ }{}^{w }q_{ }^{ }$. Let $H(x,y)$ be a 2-A-formula such that for any $\alpha \in p\left(\mathfrak{M}\right)$, $H(\mathfrak{M},\alpha )\cap q\left(\mathfrak{M}\right)\ne \varnothing$ and $\neg H(\mathfrak{M},\alpha )\cap q\left(\mathfrak{M}\right)\ne \varnothing$.

If the depth of H is equal to zero, then H should be constant on $p\left(\mathfrak{M}\right)$, which is impossible. Next we show that the depth of H on the 1-type p cannot exceed 1.

By Proposition 2, the depth of H on some convex formula $\mathsf{\Theta }\in p$ is equal to k. Towards a contradiction suppose that k is at least 2. Then there are distinct equivalence relations $E_1(x,y)$ and $E_2(x,y)$ as in the definition of the depth of a formula. Denote $H^2(x,\alpha ):=\exists z(E_2(z,\alpha )\wedge H(x,z))$ and $H^1(x,\alpha ):=\exists z(E_1(z,\alpha )\wedge H(x,z))$. Then we can derive the formula $H^2(x,\alpha )\wedge \neg H^1(x,\alpha )$ which has a non-empty intersection with $q\left(\mathfrak{M}\right)$. There are two elements ${\gamma }_1,{\gamma }_2\in q\left(\mathfrak{M}\right)$ such that $\mathfrak{M}\vDash H^1({\gamma }_1,\alpha )\wedge \neg H^2({\gamma }_2,\alpha )$. Thus, ${\gamma }_1<\left(\neg H^1(\mathfrak{M},\alpha )\cap H^2(\mathfrak{M},\alpha )\right)<{\gamma }_2$. This means that $p \cancel{\perp }{}_{ }{}^{a }q_{,}^{ }$ which is a contradiction. □

Theorem 9.

Let T be a weakly o-minimal theory having less than $2^{{\aleph }_0}$ countable models. Let each type p over a finite set be non-social (equivalently, quasisolitary). Let Γ be an infinite set of pairwise non-weakly orthogonal and almost orthogonal non-isolated types over some finite subset A of an ${\left|A\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$. Then ${\aleph }_0\le I(T,{\aleph }_0)<2^{{\aleph }_0}$.

Proof of Theorem 9.

Until the end of the proof let the theory T be small, since otherwise it has $2^{{\aleph }_0}$ countable models.

Let $\Gamma ={\left\{p_i\right\}}_{i<\omega }$. Without loss of generality, we suppose that all the $p_i$ are either irrational or quasirational to the right. For each $i<\omega$, let ${\phi }_i(x,y)$ be the greatest convex to the left A-formula on $p_i$. Denote ${\epsilon }_i(x,y):={\phi }_i(x,y)\vee {\phi }_i(y,x)$. Then the formula ${\epsilon }_i$ is a convex equivalence relation of the set of all realizations of $p_i$. Let for all distinct $i,j<\omega$$H_{i,j}$ be the A-formulas from Definition 2 guaranteeing that $p_i \cancel{\perp }{}_{ }{}^{w }{p}_j_{.}^{ }$ We can transform each $H_{i,j}$ so that $H_{i,j}(\mathfrak{M},{\alpha }_i)$ is convex and $p_j{\left(\mathfrak{M}\right)}^{-}\subseteq H_{i,j}(\mathfrak{M},{\alpha }_i)$ for each ${\alpha }_i\in p_i\left(\mathfrak{M}\right)$.

It is impossible for ${\alpha }_i\in p_i\left(\mathfrak{M}\right),{\gamma }_1,{\gamma }_2\in p_j\left(\mathfrak{M}\right)$ to be such that ${\gamma }_1\in H_{i,j}(\mathfrak{M},{\alpha }_i)$, ${\gamma }_2\notin H_{i,j}(\mathfrak{M},{\alpha }_i)$ and at the same time ${\epsilon }_j({\gamma }_2,{\gamma }_1)$. Indeed, otherwise the set $\left({\epsilon }_j(\mathfrak{M},{\gamma }_1)\cap H(\mathfrak{M},{\alpha }_i)\right)⊊p_j\left(\mathfrak{M}\right)$ would be $(\left\{{\alpha }_i\right\}\cup A)$-definable by the formula $\exists y({\epsilon }_j(x,y)\wedge H_{i,j}(x,{\alpha }_i)\wedge \neg H(y,{\alpha }_i))$, which is impossible since $p_i{\perp }^a{p}_j$.

Now fix an arbitrary $k<\omega$ and a realization ${\alpha }_k$ of $p_k$, and let ${\mathfrak{M}}_k$ be the prime model of T over $\left\{{\alpha }_k\right\}\cup A$. Let $p_k^{\prime }(x,{\alpha }_k):=p_k\left(x\right)\cup \{x<{\epsilon }_k(x,{\alpha }_k)\}$. Suppose there is ${\beta }_k\in p^{\prime }({\mathfrak{M}}_k,{\alpha }_k)$, that is ${\beta }_k\in p_k\left({\mathfrak{M}}_k\right)$ such that ${\beta }_k<{\epsilon }_k({\mathfrak{M}}_k,{\alpha }_k)$. Since ${\mathfrak{M}}_k$ is prime over ${\alpha }_k$, there is an A-formula $\psi (x,{\alpha }_k)$ such that ${\beta }_k\in \psi ({\mathfrak{M}}_k,{\alpha }_k)\subseteq p_k^{\prime }({\mathfrak{M}}_k,{\alpha }_k)$ and $\psi (x,{\alpha }_k)$ isolates a type over ${\alpha }_k\cup A$. Then $\exists x^{\prime }(\psi (x^{\prime },{\alpha }_k)\wedge x^{\prime }\le x\le {\alpha }_k)$ is a convex to the left on $p_k$ formula greater than ${\phi }_k$, which is impossible. Then there can be no such ${\beta }_k\in p_k^{\prime }\left({\mathfrak{M}}_i\right)$, and $p_k^{\prime }$ should be omitted in ${\mathfrak{M}}_k$. Then ${\epsilon }_i{({\mathfrak{M}}_k,{\alpha }_k)}^{-}=p_k{\left({\mathfrak{M}}_k\right)}^{-}$. Next we show that a similar property does not hold for $p_j\left({\mathfrak{M}}_k\right)$.

Towards a contradiction suppose that there are $j\ne k$ and ${\beta }_j\in p_j\left({\mathfrak{M}}_k\right)\cap H_{k,j}({\mathfrak{M}}_k,{\alpha }_k)$. Then $tp({\beta }_j/A{\alpha }_k)$ is isolated, let $\mu (x,{\alpha }_k)$ be the A-formula isolating this type. Then $\mu ({\mathfrak{M}}_k,{\alpha }_k)\subseteq H_{k,j}({\mathfrak{M}}_k,{\alpha }_k)\cap p\left({\mathfrak{M}}_k\right)$. Note that for any ${\left|A\right|}^{+}$-saturated model $\mathfrak{M}\vDash T$ with $A\cup \left\{{\alpha }_k\right\}\subseteq M$, $\mu (\mathfrak{M},{\alpha }_i)\ne H_{k,j}(\mathfrak{M},{\alpha }_k)\cap p\left(\mathfrak{M}\right)$. Then the formula $\mu (x,y)$ guarantees that $p_k \cancel{\perp }{}_{ }{}^{a }{p}_j_{,}^{ }$ which is a contradiction. Therefore, we obtained that $p_j\left({\mathfrak{M}}_k\right)\cap H_{k,j}({\mathfrak{M}}_k,{\alpha }_k)=\varnothing$.

(a) If all the $p_i$ are irrational, analogically to the previous part, we obtain $p_j\left({\mathfrak{M}}_k\right)\cap H({\mathfrak{M}}_k,{\alpha }_k)=\varnothing$ for each $j\ne k$. Then $p_j\left({\mathfrak{M}}_k\right)=\varnothing$ and ${\mathfrak{M}}_k$ omits every $p_j$, $j\ne i$. Fix for every $i<\omega$ a prime model ${\mathfrak{M}}_i$ over the set A and an arbitrary realization of $p_i$. Then each ${\mathfrak{M}}_i$ realizes $p_i$ and omits all the $p_j$ for $j\ne i$. Then for distinct $i_1,i_2<\omega$, ${\mathfrak{M}}_{i_1} \ncong {\mathfrak{M}}_{i_2}$, and $I(T,{\aleph }_0)\ge {\aleph }_0$.

(b) Suppose that all the $p_i$ are quasirational to the right. For each $i<\omega$ let $U_i\left(x\right)$ be the A-formula from Definition 7 guaranteeing quasirationality of $p_i$.

The $(A\cup \left\{{\alpha }_k\right\})$-formula $\neg H_{k,j}(x,{\alpha }_k)\wedge U_j\left(x\right)$ defines the set $p_j\left(\mathfrak{M}\right)\setminus H_{k,j}(\mathfrak{M},{\alpha }_k)$. Since the theory T is small, this formula has an isolated over $(A\cup \left\{{\alpha }_k\right\})$ subformula, which has to be realized in $p_j\left({\mathfrak{M}}_k\right)$. Therefore, $p_j\left({\mathfrak{M}}_k\right)=p_j\left({\mathfrak{M}}_k\right)\setminus H({\mathfrak{M}}_k,{\alpha }_k)\ne \varnothing$.

Thereby, the set $p_k\left({\mathfrak{M}}_k\right)$ contains the least ${\epsilon }_k$-class, namely, ${\epsilon }_k({\mathfrak{M}}_k,{\alpha }_k)$, while for each $j\ne k$, the set $p_j\left({\mathfrak{M}}_k\right)$ has no least ${\epsilon }_j$-class. Since the index k was arbitrary, for each $i<\omega$, we can fix a prime model ${\mathfrak{M}}_i$ over the set A and some realization of $p_i$ with a least ${\epsilon }_i$-class and no least ${\epsilon }_j$-class for each $j\ne i$. Then, T has at least ${\aleph }_0$ countable models. □

5. Conclusions

The article applied L. Mayer’s approach to a wider range of theories. It considered the case of an infinite number of countable, non-isomorphic models of a small, weakly o-minimal theory with a non-maximal number of countable models. Therefore, in order to solve the problem of the countable spectrum of weakly o-minimal theories, the case of the maximum number of countable models remains to be described.