On the Kaiser Class of a Jonsson Theory

Abstract

This work introduces new tools to extend the research framework for Jonsson theories, which form a special subclass of inductive theories. Specifically, we introduce the concept of the Kaiser class and define a new equivalence relation for the study of Jonsson theories, which refines the relation of cosemanticness of the theories. This paper presents an analysis of the main properties of the Kaiser class and $K_T$-equivalent Jonsson theories, addresses the axiomatization of the Kaiser class, and formulates criteria for the perfectness of a Jonsson theory and the $cl$-normality of a perfect Jonsson theory. Furthermore, we thoroughly examine the algebraic structure of the cosemanticness class of a given Jonsson theory, demonstrate that a distributive lattice can be introduced on this class, and construct specialized sublattices of this lattice. The results obtained can be applied in the study of Jonsson theories and model-theoretic questions with the application of the methods of lattice theory and Universal Algebra.

1. Introduction

This work belongs to the field of Model Theory, one of the dynamically developing branches of Mathematical Logic. Contemporary topics from this discipline have many intersections with the modern questions that originate from the tasks of the following areas: Abstract Algebra, Universal Algebra, Algebraic Geometry, and topology. Model Theory studies formal languages and their interpretations, or models, with a focus on the interplay between syntactic and semantic structures.

Historically, Model Theory has evolved along two distinct trajectories, often referred to as the “western” and “eastern” schools. This division is rooted in the geographical locations of its pioneers: Alfred Tarski, who resided on the West Coast of the United States, and Abraham Robinson, who was based on the East Coast. The western approach emphasizes problems in Mathematical Analysis and Algebraic Geometry, typically assuming the completeness of the formal languages under consideration. In contrast, the eastern approach is grounded in Universal Algebra and Abstract Algebra, often dealing with incomplete theories and their applications. This research aligns with the Eastern tradition, focusing on the study of incomplete theories, particularly the class of Jonsson theories.

Jonsson theories form a specific subclass of inductive theories, characterized by three key model-theoretic properties: the existence of infinite models, the amalgamation property, and the joint embedding property. According to the definition, Jonsson theories are generally not complete, making their study both more challenging and more general. The absence of elementary equivalence between models of incomplete theories introduces significant complexity, which is why the class of inductive theories—and Jonsson theories in particular—has become a central object of study. Examples of Jonsson theories include the theory of all groups, the theory of abelian groups, the theory of fields of fixed characteristic, the theory of linear orders, and many others. While these are typically incomplete, there are also complete Jonsson theories, such as the theory of divisible abelian groups, the theory of algebraically closed fields of fixed characteristic, and the theory of dense linear orders without endpoints. This duality makes the class of Jonsson theories both versatile and rich in structure, offering a fertile ground for model-theoretic research.

Contemporary research on Jonsson theories has been significantly advanced by the Karaganda School of Model Theory, with key contributions by T.G. Mustafin. Later on, his scientific research in the field of Jonsson theories was continued by representatives of the Karaganda School of Model Theory. The main results concerning Jonsson theories, including those established by the authors mentioned above, as well as the contributions of this work, are detailed in Section 1 of this paper.

In recent years, members of the Karaganda School of Model Theory have developed a specialized theoretical framework for studying Jonsson theories. This framework incorporates techniques rooted in the theory of companions, originally introduced by A. Robinson. One such technique is the semantic method, first proposed by T.G. Mustafin. The semantic method focuses on analyzing a Jonsson theory by studying its semantic model, which is an ${\omega }^{+}$-universal and ${\omega }^{+}$-homogeneous model of the theory. Since the same structure can serve as a semantic model for multiple Jonsson theories in a given language, this method allows us to describe entire classes of Jonsson theories that have the same semantic model. Moreover, this approach enables the study of algebraic structures through their theories in a fixed countable first-order language L.

The semantic model $C_T$ of a Jonsson theory T serves as the primary semantic invariant for a class of cosemantic Jonsson theories. To study the syntactic properties of this model, we consider its elementary theory, which is called the center of the given theory T. To understand the semantic properties of $C_T$, we examine special definable subsets of the model, such as Jonsson sets and almost Jonsson sets. The study of these subsets involves a closure operator that induces a topology on the model. Common examples of such operators include the definable closure operator $dcl\left(X\right)$ and the algebraic closure operator $acl\left(X\right)$, where X is a Jonsson or almost Jonsson set. For example, considering the theory $T_F$ of fields of the fixed characteristic $p\ge 0$, which is a Jonsson theory, and the theory $T_F^{*}$ of algebraically closed fields of the same characteristic, which is the center of $T_F$, it is well-known that $cl=acl$. In this paper, we do not focus on topological properties of the definable sets but deal with the models that are the closures of almost Jonsson sets, regarding some given pregeometry $\tau$ with some abstract closure operator $cl$. Jonsson sets and their generalization—almost Jonsson sets—act as a kind of “basis” for studying the semantic properties of the model, and their closures form a special subclass of Jonsson theories. Specifically, the closure of a Jonsson set is an existentially closed model of the theory, while the class of closures of almost Jonsson sets relates to the Kaiser class of the Jonsson theory. More precisely, the Kaiser class $K_T$ of the Jonsson theory T, which consists of all models of T whose Kaiser hull is a Jonsson theory. The Kaiser class $K_T$ of a Jonsson theory T is proposed as a new, more sophisticated semantic invariant of the Jonsson theory, becoming a carrier of the most essential Jonsson properties of the class of models of T.

The problem of describing and, in particular, axiomatizing the class of existentially closed models of a theory has long been an open and challenging question in Model Theory. Similarly, the axiomatization of the Kaiser class of a Jonsson theory becomes a difficult problem. Solving this problem would significantly enhance our ability to describe the syntactic and semantic properties of Jonsson theories. It is known that the classes of existentially closed models of cosemantic Jonsson theories coincide. In this work, we focus on a narrower class of Jonsson theories whose Kaiser classes coincide. We describe the key properties of the Kaiser class and use this concept to formulate necessary and/or sufficient conditions for characterizing special subclasses of Jonsson theories, such as perfect and $cl$-normal Jonsson theories. Section 2 of this paper is dedicated to these results, including the problem of axiomatization of the Kaiser class.

In addition, this work extends the technical framework of the semantic method by studying the algebraic properties of classes of theories that have the same semantic model—that is, cosemantic Jonsson theories. The problem of algebraizing theories is a central concern in modern Model Theory, Mathematical Logic, and Universal Algebra. Key works in this direction [1,2,3,4,5] have explored the closure of classes of L-sentences under certain algebraic properties, such as closure under direct products, Horn conditions, and operations defined between theories. However, these results primarily concern complete theories. The novelty of our research lies in applying methods from lattice theory to Jonsson theories, which, as mentioned earlier, are generally incomplete. In this context, we introduce the concept of the Kaiser class $K_T$ of a theory T and $K_T$-equivalence of theories as new key tools in the framework of Jonsson Model Theory. Section 3 of this paper describes the lattice of cosemantic Jonsson theories and its special sublattices. These results extend our earlier work [6], where we introduced certain algebraic structures defined on the class of Jonsson theories.

Let us suggest that our current work is not merely a novelty in terms of introducing new notions; rather, we propose that the concepts of $cl$-normality and $K_T$-equivalence offer a new method and perspective within the classical framework for studying incomplete inductive theories, such as Jonsson theories.

A single observation suffices to illustrate the significance of this approach: numerous unresolved problems persist concerning the axiomatizability of concrete properties of fixed classes of models for such theories. Consequently, a variety of related issues naturally arise—among them, the existence and classification of various types of companions, as well as their connections to long-standing open questions in classical Model Theory. More specifically, the present study is devoted to a foundational analysis of the model-theoretic properties of the Kaiser class of a Jonsson theory, including the theory of groups. In our future work, we aim to determine which subclasses of the Kaiser class of the theory of groups correspond exactly to the class of simple groups.

One such problem is linked to the still unproven Zilber–Cherlin conjecture. A particular task associated with this conjecture concerns the well-known hypothesis that existentially closed groups are simple. However, the class of existentially closed models of the theory of all groups is, in fact, a subclass of the Kaiser class of that theory. In our view, the notion of $cl$-normality is a potentially crucial tool in approaching this problem.

This paper is organized as follows: an introduction, three main sections, and a bibliography. Below, we outline the conventions and notation used throughout the text.

We work in a countable first-order language L. By an L-theory T, we mean any consistent set of sentences in L closed under logical consequence. In case when some set S of sentences is not closed under logical consequence but we call it a theory, we mean the deduction closure of S. The class of all L-structures that are models of T is denoted by $Mod\left(T\right)$. If $M\in Mod\left(T\right)$, we write $M\vDash T$ to indicate that all sentences of T are true in M. The axioms $Ax\left(T\right)$ of T are the set of L-sentences such that for any model M of T, $M\vDash Ax\left(T\right)$, and vice versa. Two L-theories $T_1$ and $T_2$ are logically equivalent if $Mod\left(T_1\right)=Mod\left(T_2\right)$, and in this case we write $T_1=T_2$. For a class K of L-structures, $K\vDash T$ means that every structure $A\in K$ satisfies $A\vDash T$. By $T_{\forall }$, we mean the set of all universal consequences of the theory T; similarly, $T_{\forall \exists }$ is the set of all universal–existential sentences deduced in T. The Kaiser hull of K, denoted by $T{h}_{\forall \exists }\left(K\right)$, is the set of all $\forall \exists$-sentences in L true in every structure of K. We also denote the Kaiser hull of K by $T^0\left(K\right)$. For any two L-structures A and B, we write $A{\prec }_{1}B$ if $A\subseteq B$, and for any universal L-formula $\phi (\overline{x}$ and element $\overline{a}\in A$, if $A\vDash \phi \left(\overline{a}\right)$, then $B\vDash \phi \left(\overline{a}\right)$.

2. Basic Facts on Jonsson Theories

In this section, we present the necessary notions and facts concerning Jonsson theories. We start by describing some classical model-theoretic concepts, such as the amalgamation property, the joint embedding property, and the existentially closed model.

We start with the concepts of amalgamation and joint embedding properties. Originally arising from Algebra, the properties of amalgamation (AP) and joint embedding (JEP) have become fundamental in both classical and modern Model Theory. Amalgamation allows the combination of models with a common submodel, preserving certain properties in the larger model, while joint embedding ensures that two models can be embedded into a third model, revealing their interrelationships. Recall the definitions.

Let K be a class of L-structures.

Definition 1

(p. 80, [7]). K admits the amalgamation property (AP), if for any models A, $B_1$, $B_2\in K$, and isomorphic embeddings $f_1:A\to B_1$, $f_2:A\to B_2$, there are $M\in K$ and isomorphic embeddings $g_1:B_1\to M$, $g_2:B_2\to M$, such that $g_1\circ f_1=g_2\circ f_2$.

Definition 2

(p. 80, [7]). The class K admits the joint embedding property (JEP), if for any models $A,B\in K$, there exists a model $M\in K$ and isomorphic embeddings $f:A\to M$, $g:B\to M$.

We say that an L-theory admits AP or JEP, if $Mod\left(T\right)$ admits AP or JEP, correspondingly.

In Model Theory, these properties are crucial in the study of categorical structures, stability theory, and classification theory, providing key insights into the interactions between models, model completeness, quantifier elimination, and the transfer of properties across models.

Given that we are working within the framework of inductive theories, the study of amalgamation and joint embedding naturally leads to the consideration of existentially closed (e.c.) models. A key result in Model Theory states that for any model A of an inductive theory T, there exists an e.c. model M of T such that A embeds into M. This result is crucial for understanding the structure of models in inductive theories, as e.c. models provide a canonical way to extend any given model within the theory. Let us recall the definition of the existentially closed model.

Definition 3

(p. 97, [7]). A model M of a theory T is said to be existentially closed if for every embedding $M\to N$ into another model N of T and every existential formula $\phi \left(x\right)$ with parameters from M that is satisfiable in N, it is also satisfiable in M.

There is also another definition of existentially closed models proposed by H. Simmons, which is equivalent to Definition 3.

Definition 4

([8]). A structure A is called an existentially closed model of T, if $A\vDash T$ and, for any model B of T,

$$A\subseteq B\Rightarrow A{\prec }_{1}B.$$

In this paper, we use the results of H. Simmons on existentially closed models, and this is why we prefer to follow Definition 4 for the concept of existentially closed model.

We now recall some classical results on existentially closed models, which are essential for the present study. The first one allows us to determine whether a model of a theory T is an e.c. model.

Theorem 1

([8]). If $A{\prec }_{1}B$, where $A\vDash T$ and B is an existentially closed model of T, then A is also an existentially closed model of T.

We now turn to Jonsson theories, which are the central focus of this study. As it was mentioned in the introduction, these theories are fundamental in the context of Model Theory, particularly within the framework of inductive theories. The following sections address the key properties of Jonsson theories and their relevance to the present research.

Definition 5

(p. 80, [7]). A theory T is called a Jonsson theory, if the following conditions hold:

(1) 

T has at least one infinite model,

(2) 

T is an inductive theory (or $\forall \exists$-axiomatizable, which is equivalent),

(3) 

T admits JEP,

(4) 

T admits AP.

The relevance of studying Jonsson theories is demonstrated by a significant collection of classical algebraic examples. These examples show that such algebras can be studied not only through traditional algebraic methods but also within the framework of Jonsson theory. The following examples of Jonsson theories illustrate the applicability of this approach:

Example 1.

(1) 

The theory of all groups;

(2) 

The theory of all abelian groups;

(3) 

The theory of all Boolean algebras;

(4) 

The theory of all linear orders;

(5) 

The theory of all fields of characteristic p, where p is zero or a prime number;

(6) 

The theory of ordered fields;

(7) 

The theory of R-modules;

(8) 

The theory of differential fields of characteristic 0;

(9) 

The theory of differentially closed fields of characteristic 0;

(10) 

The theory of differentially perfect fields of characteristic p;

(11) 

The theory of differentially closed fields of characteristic p.

It follows from the definition that Jonsson theories are, in general, not complete. As a result, for a Jonsson theory T, we are not able to compare models in terms of elementary equivalence within the class of all models of T. However, such a comparison is possible within an important subclass—namely, the class of existentially closed models of T. The following classical result due to W. Hodges enables this.

The following result was obtained by W. Hodges.

Theorem 2

([9]). Suppose T is an L-theory, and let T admit JEP. Let A and B be an existentially closed model of T. Then, each $\forall \exists$-sentence that is true in A is true in B as well.

According to the result, any two existentially closed models of T are elementarily equivalent with respect to a restricted fragment of the language—specifically, the fragment consisting of universal–existential sentences. Nonetheless, as will be seen later, this restriction to universal–existential sentences plays an essential role in the description of the Kaiser class of models of the theory T.

The foundation for the study of Jonsson theories was laid by T. G. Mustafin, who initially defined the semantic model of a Jonsson theory under the assumption of an additional axiom on the existence of a strongly inaccessible cardinal. Later, Ye. T. Mustafin reformulated this definition to eliminate the need for this assumption:

Definition 6

(Ye. T. Mustafin [10]). A model $C_T$ of a Jonsson theory T is called a semantic model of this theory, if $|C_T|=2^{\omega }$ and $C_T$ is ${\omega }^{+}$-universal ${\omega }^{+}$-homogeneous model of T.

The semantic model plays a central role in the study of a fixed Jonsson theory, as it serves as a distinctive semantic invariant for the theory. This is demonstrated by the following theorem. The emergence of the notion of a semantic model for the study of an arbitrary Jonsson theory is of twofold importance. First, the semantic model serves as a faithful analogue of the monster model in the context of complete theories. Second, the elementary theory of this model—referred to as the center—constitutes a syntactic invariant that reflects the internal content of the given Jonsson theory.

Theorem 3

(Ye. T. Mustafin [10]). An inductive theory T is Jonsson if it has a ${\omega }^{+}$-universal ${\omega }^{+}$-homogeneous model (which is its semantic model).

In addition to its importance for the study of inductive theories, the class of existentially closed models holds a particularly special significance in the context of Jonsson theories, as demonstrated by the following theorem.

Theorem 4

(T. G. Mustafin [11]). For any Jonsson theory T, $C_T$ is an existentially closed model of T, i.e., $C_T\in E_T$.

As mentioned in the introduction, much of Model Theory focuses on complete theories, and, consequently, the tools for studying complete theories are both more extensive and better developed. In the study of Jonsson theories, we occasionally apply the techniques used in the study of complete theories. This is facilitated by the following concept.

Definition 7

(T. G. Mustafin [11]). Let T be a Jonsson theory. Then, theory $T^{*}=Th\left(C_T\right)$ is called a center of T.

Note that by this definition, the center of a Jonsson theory is a complete theory, which allows us to transfer the syntactic properties of the center to the Jonsson theory itself, which may, in general, be incomplete. In cases where we apply the companion theory technique, the following fact, derived from Definition 7, proves particularly useful. It establishes a connection between the classes of models of the given Jonsson theory and its center.

Corollary 1

(T. G. Mustafin [11]). Let T be a Jonsson theory, $T^{*}$ be its center. Then, T and $T^{*}$ are mutually model consistent, that is $T_{\forall }=T_{\forall }^{*}$.

Moreover, the following well-known result shows that the class of e.c. models of a Jonsson theory T preserve the Jonsson properties of T when constructing the Kaiser hull of the class of e.c. models of the given Jonsson theory.

Theorem 5

(A. R. Yeshkeyev [11]). Let T be a Jonsson theory, and $E_T$ be the class of existentially closed models of T. Then, $T{h}_{\forall \exists }\left(E_T\right)=E_T^0$ is a Jonsson theory.

Note that by virtue of Theorem 2, Theorem 5 stays true for any subclass $K\subseteq E_T$, i.e., $K^0$ is also a Jonsson theory.

When studying the properties of Jonsson theory through the description of its semantic model and center, particular subclasses of Jonsson theories emerge. One such subclass is the class of perfect Jonsson theories. We now provide the definition of a perfect Jonsson theory.

Definition 8

([10]). A Jonsson theory T is called perfect, if $C_T$ is ${\omega }^{+}$-saturated.

The examples of perfect Jonsson theory are represented by points 2–5 and 8–10 in Example 1.

In the study of perfect Jonsson theories, the theory of companions is used. Let us recall the definition of a companion.

Definition 9

([7]). Let T be a theory in the language L. An L-theory $T^{\#}$ is called a companion of T, if the following conditions hold:

(1) 

$T_{\forall }=T_{\forall }^{\#}$;

(2) 

whenever $T_{\forall }=T_{\forall }^{\prime }$, ${\left(T^{\prime }\right)}^{\#}=T^{\#}$;

(3) 

$T_{\forall \exists }\subseteq T^{\#}$.

A classic example of a companion of a theory is the model companion. Note that the existence of a model companion for an arbitrary theory is not guaranteed, and the problem of a complete description of the theories for which this is possible remains an open question. We now provide the definition of a model companion.

Definition 10

([7]). Let T be a theory in L. A theory $T^{MC}$ is called a model companion of T, if $T^{MC}$ is a companion of T and $T^{MC}$ is a model complete theory.

The effectiveness of the technique of companions is clearly demonstrated by the following theorem.

Theorem 6

(A. R. Yeshkeyev [11]). Let T be a Jonsson theory. Then, the following conditions are equivalent:

(1) 

T is a perfect Jonsson theory;

(2) 

$T^{*}$ is a model companion of T;

(3) 

$Mod\left(T^{*}\right)=E_T$.

When describing the properties of a Jonsson theory, we are essentially examining the properties of its semantic model. However, different Jonsson theories may have the same semantic model. Thus, we are not describing a single theory, but an entire class of theories that share a common semantic model. This is the essence of the semantic approach in the study of Jonsson theories. The central concept in this context is the definition of cosemantic Jonsson theories.

Definition 11

(T. G. Mustafin [11]). Let $T_1$ and $T_2$ be Jonsson L-theories, $C_1$ and $C_2$ be their semantic models, correspondingly. $T_1$ and $T_2$ are said to be cosemantic (denoted by “$T_1\bowtie T_2$”), if $C_1=C_2$.

In the study of Jonsson theories, cosemanticness is primarily used as a tool for comparing Jonsson theories from the perspective of their semantic invariants, that is, their semantic models. Many important model-theoretic properties of cosemantic Jonsson theories coincide, which allows us to work not just with a single Jonsson theory, but with an entire class of theories that are cosemantic to it. This approach is directly related to the so-called semantic method proposed by T. G. Mustafin and successfully applied in the study of Jonsson and, more generally, inductive theories.

Let us provide examples of cosemantic Jonsson theories.

Example 2.

(1) 

The theory of abelian groups and the theory of divisible abelian groups are cosemantic;

(2) 

The theory of fields of characteristic p, where $p\ge 0$, is cosemantic to the theory of algebraically closed fields of the same characteristic;

(3) 

The theory of differential fields of characteristic 0 and the theory of differentially closed fields of the same characteristic;

(4) 

The theory of of differentially perfect fields of characteristic p and the theory of differentially closed fields of the same characteristic.

Examples of noncosemantic Jonsson theories are the theory of groups and the theory of abelian groups. Note that the first one is not a perfect Jonsson theory, while the latter is.

It is necessary to note that the cosemanticness relation is an equivalence relation. Let us briefly show the following:

(1)

For any Jonsson theory T, it is clear that $C_T=C_T$, so $T\bowtie T$;

(2)

If $T_1\bowtie T_2$ that is $C_{T_1}=C_{T_2}$ then $C_{T_2}=C_{T_1}$ and $T_2\bowtie T_1$;

(3)

Whenever $T_1\bowtie T_2$ and $T_2\bowtie T_3$, it is true that $T_1\bowtie T_3$, since $C_{T_1}=C_{T_2}=C_{T_3}$.

Thus, cosemanticness is a reflexive, symmetric, and transitive binary relation.

The study of cosemantic Jonsson theories was also conducted by Ye. T. Mustafin. In his work [10], he outlined several fundamental properties of cosemantic Jonsson theories, and the following is one of them.

Proposition 1

([10]). If T is a Jonsson theory and $T^{\prime }$ is an inductive theory such that $T_{\forall }=T_{\forall }^{\prime }$, then $T^{\prime }$ is a Jonsson theory that is cosemantic to T.

Proposition 1 naturally implies the following statement:

Proposition 2

([6]). Let $T_1$ and $T_2$ be Jonsson theories. Then, $T_1\bowtie T_2$ iff $E_{T_1}=E_{T_2}$.

The following fact was obtained by the first author of this paper and allows one to identify the cosemantic Jonsson theories:

Theorem 7

([11]). Let $T_1$ and $T_2$ be two Jonsson theories and let $C_{T_1}\vDash T_2$, $C_{T_2}\vDash T_1$. Then, $T_1\bowtie T_2$.

The following two theorems also serve as useful tools in working with cosemantic Jonsson theories, as they allow the construction of extensions of the given Jonsson theory that will be cosemantic to it. Despite their simplicity, these theorems hold significance for our research. Theorem 8 allows us to identify whether an inductive extension of a Jonsson theory T is cosemantic to T. We call such theories cosemantic extensions of T.

Theorem 8

([6]). Let T be a Jonsson L-theory, $C_T$ be its semantic model, and $T^{\prime }$ be a theory of the same language such that $T^{\prime }$ is inductive, $T\subseteq T^{\prime }$, and $C_T\vDash T^{\prime }$. Then $T^{\prime }$ is a Jonsson theory that is cosemantic to T.

The next theorem is a consequence of Theorem 8 and also shows the way to construct a cosemantic extension of T, but using only existential sentences.

Theorem 9.

Let T be a Jonsson L-theory, $C_T$ be its semantic model, and Σ be a set of existential L-sentences such that $T\cup \left\{\psi \right\}$ is consistent for any $\psi \in \mathsf{\Sigma }$. Then, $T\cup \mathsf{\Sigma }$ is a Jonsson theory that is cosemantic to T.

Proof. 

According to Theorem 8, it is sufficient for $T\cup \mathsf{\Sigma }$ to be satisfied in $C_T$ to be a cosemantic to T. Assume the contrary, let $C_T$ not be a model of $T\cup \mathsf{\Sigma }$. It means that $C_T⊭\mathsf{\Sigma }$. Then, there is an existential sentence $\phi \in \mathsf{\Sigma }$ such that $C_T⊭\phi$. Consequently, $C_T\vDash \neg \phi$, where $\neg \phi$ is equivalent to some universal L-sentence. $C_T$ is an existential model of T due to Theorem 4, and by Theorem 2, for any existentially closed model M of T, $M\vDash \neg \phi$. As it was mentioned, for any model $A\in Mod\left(T\right)$, A is embedded into some existentially closed model M of T, since T is inductive. Then, from $M\vDash \neg \phi$, it follows that $A\vDash \neg \phi$. Hence, any model of T satisfies $\neg \phi$, which means that $T⊢\neg \phi$, which is a contradiction. Therefore, $C_T\vDash \phi$. We can state this for all sentences $\phi \in \mathsf{\Sigma }$ using transfinite induction. Thus, applying Theorem 8, we obtain that $T\cup \left\{\psi \right\}$ is cosemantic to T. □

Let ⌀ be an empty set of sentences of the language L. By definition of an L-theory, ⌀ is a theory in L. The class of models of ⌀ is represented by all L-structures. From [10], it is known that ⌀ is a Jonsson theory. Then, Theorem 9 has the following interesting corollary, which gives us a new specific example of a Jonsson theory for our further research on cosemantic theories:

Corollary 2.

Any existentially axiomatized L-theory is a Jonsson theory cosemantic to ⌀.

Thus, all existentially axiomatized theories are cosemantic to each other.

Now, we apply Theorems 8 and 9 to derive the following results, which are critical for obtaining further conclusions in the next section. These results enable the construction of necessary extensions and provide the foundational tools for the subsequent analysis of the considered Jonsson theories.

Theorem 10.

Let T be a Jonsson L-theory and $T_{\infty }$ be an L-theory such that

$$Mod\left(T_{\infty }\right)=\{A\mid A\in Mod\left(T\right)\mathrm{and}\left|A\right|\ge \omega \}.$$

Then $T_{\infty }$ is a Jonsson theory cosemantic to T.

We demonstrate two alternative proofs for this fact.

Proof 1.

Let us show that $T_{\infty }$ using the definition of a Jonsson theory (Definition 5).

(1)

Obviously, $T_{\infty }$ has infinite models.

(2)

The class $Mod\left(T_{\infty }\right)$ is closed under the unions of chains. Firstly, note that $Mod\left(T_{\infty }\right)\subseteq Mod\left(T\right)$. Let ${\left\{A_i\right\}}_{i\in I}$ be a chain of models of $T_{\infty }$, then ${\bigcup }_{i\in I}{A}_i=A\in Mod\left(T\right)$, since T is inductive. It is clear that A is infinite, then $A\in Mod\left(T_{\infty }\right)$. Therefore, $T_{\infty }$ is an inductive theory.

(3)

Let $A,B\in Mod\left(T_{\infty }\right)$. Then, there is a model $M\in Mod\left(T\right)$ such that A and B are embedded into M. A and B are infinite models; therefore, M is also infinite and $M\in Mod\left(T_{\infty }\right)$. Therefore, $T_{\infty }$ admits JEP.

(4)

Let $A,B,C\in Mod\left(T_{\infty }\right)$ and f and g be isomorphic embeddings of A and B into C, respectively. Then, there exists $D\in Mod\left(T\right)$ and isomorphic embeddings $f_1$ and $g_1$ such that $f:B\to D$ and $C\to D$. $A,B,C$ are infinite, then D is also infinite and, consequently, $D\in Mod\left(T_{\infty }\right)$. Therefore, $T_{\infty }$ admits AP. So $T_{\infty }$ is a Jonsson theory. Since $T_{\infty }$ is an inductive theory, $T\subseteq T_{\infty }$ and $C_T\vDash T_{\infty }$, by Theorem 8, $T_{\infty }\bowtie T$.

□

Proof 2.

It is clear that

$$T_{\infty }=T\cup {\mathsf{\Phi }}_{\infty },$$

where ${\mathsf{\Phi }}_{\infty }$ is the following set of L-sentences:

$${\mathsf{\Phi }}_{\infty }=\left\{\exists x_0\dots x_n\underset{i<j\le n}{\bigwedge }{x}_i\ne x_j\mid n\in \omega \right\},$$

that is ${\mathsf{\Phi }}_{\infty }$ declares the infinity of a structure. All sentences $\phi \in {\mathsf{\Phi }}_{\infty }$ are existential, and then $T_{\infty }=T\cup {\mathsf{\Phi }}_{\infty }$ is a Jonsson theory cosemantic to T according to Theorem 9. □

Another canonical tool of the semantic method is the concept of the Jonsson spectrum and cosemanticness of L-structures. Let us recall the definition of the Jonsson spectrum of a class of L-structures.

Definition 12

([12]). Let K be a class of L-structures. A Jonsson spectrum $JSp\left(K\right)$ of K is the following set of theories

$$JSp\left(K\right)=\left\{T\right|T\mathit{is}\mathrm{a}\mathit{Jonsson}\mathit{theory}\mathit{and}\forall A\in KA\vDash T\}.$$

Initially, the concept of cosemanticness of L-structures was proposed by Ye Nurkhaidarov, but it was later reformulated by the first author of this article in order to clarify and expand its applicability within the context of the semantic method. Thus, the following formulation is now relevant:

Definition 13

([11]). L-structures A and B are called cosemantic (denoted by $A\bowtie B$), if $JSp\left(A\right)=JSp\left(B\right)$.

The concept of the cosemanticness of L-structures serves as a generalization of the notion of elementary equivalence within the framework of Jonsson theory of models, and allows us to encompass a broader class of Jonsson theories under consideration.

In [13], the concept of cosemanticness of L-structures was generalized to the cosemanticness of classes of L-structures. In this manner, we have the following definition:

Definition 14

([13]). Let $K_1$ and $K_2$ be some classes of L-structures. Then, $K_1$ and $K_2$ are said to be cosemantic $(K_1\bowtie K_2)$, if $JSp\left(K_1\right)=JSp\left(K_2\right)$.

For a more detailed structural description of the semantic model, we focus on describing the properties of a special class of definable subsets within the model. This method is also a useful feature of the framework used to study Jonsson theories. Moreover, in defining these subsets, we introduce a specific topology on the semantic model of the theory, which broadens the applicability of the method across various mathematical domains. In this paper, when dealing with notions that require the specification of a closure operator, we consider Jonsson theories whose semantic models admit a pregeometry of a special kind, namely, a Jonsson pregeometry.

Definition 15

([11]). Let T be a Jonsson theory, and let $X\subseteq C_T$. The pair $(X,cl)$ a Jonsson pregeometry τ, if there is a closure operator $cl:P\left(C_T\right)\to P\left(C_T\right)$ such that the following conditions hold:

(1) 

If $A\subseteq X$, then $A\subseteq cl\left(A\right)$ and $cl\left(cl\right(A\left)\right)=cl\left(A\right)$;

(2) 

If $A\subseteq B\subseteq X$, then $cl\left(A\right)\subseteq cl\left(B\right)$;

(3) 

(exchange) If $A\subseteq X$, $a,b\in X$, and $a\in cl(A\cup \{b\left\}\right)\backslash cl\left(A\right)$, then $b\in cl(A\cup \{a\left\}\right)$;

(4) 

(finite character) If $A\subseteq X$ and $a\in cl\left(A\right)$, then there is a finite $A_0\subseteq A$ such that $a\in cl\left(A_0\right)$.

In this paper, we do not place emphasis on the topological properties of these subsets, but we describe some of the properties related to them using the new concepts defined later in the article. In doing so, we propose new tools to advance the framework for studying the semantic model. Let us give the definitions of some types of described subsets.

Definition 16

(p. 297, [11]). Let T be a Jonsson theory and $C_T$ be its semantic model, $X\subseteq C$. X is said to be a Jonsson subset of $C_T$, if X is an ∃-definable set and $cl\left(X\right)=M$, where $M\in E_T$.

For the syntactic description of the properties of the semantic model of a given Jonsson theory using the concepts of Jonsson subsets, we introduce the notion of a Jonsson fragment of a Jonsson subset $X\subseteq C_T$.

Definition 17

(p. 298, [11]). The fragment of a Jonsson subset X is a theory $Fr\left(X\right)=T{h}_{\forall \exists }\left(M\right)$, where $M=cl\left(X\right)$.

Let T be a Jonsson theory, $X\subseteq C_T$, and let $cl\left(X\right)=M\in E_T$. That is, X is a Jonsson subset of $C_T$. Then, $Fr\left(X\right)=T{h}_{\forall \exists }\left(M\right)$ and, moreover, the following lemma is true:

Lemma 1

(p. 299, [11]). For any Jonsson set $X\subset C_T$, the fragment $Fr\left(X\right)$ is a Jonsson theory.

Obviously, all axioms of T are true in a semantic model of $Fr\left(X\right)$, that is $C_{Fr\left(X\right)}\in Mod\left(T\right)$, and moreover, $C_{Fr\left(X\right)}$ is existentially closed over T. It means that whenever X is a Jonsson set for T, the semantic model of $Fr\left(X\right)$ is always embedded in $C_T$ and an existentially closed submodel of $C_T$ for any Jonsson theory T. To generalize this case and refine possible situations in the context of studying Jonsson theories, we introduce the following notions:

Definition 18

([14]). Let T be a Jonsson theory, $C_T$ be its semantic model, $X\subseteq C$. X is called an almost Jonsson subset of $C_T$, if X is an ∃-definable set and $cl\left(X\right)=M$, where $M\in Mod\left(T\right)$, and $T{h}_{\forall \exists }\left(M\right)$ is a Jonsson theory.

By analogy with the concept of a Jonsson set, for an almost Jonsson subset $X\subseteq C_T$ of the theory T, we consider the fragment of X:

Definition 19

([14]). The fragment of an almost Jonsson subset X is a theory $Fr\left(X\right)=T{h}_{\forall \exists }\left(M\right)$, where $M=cl\left(X\right)$.

Based on the previously introduced concepts of Jonsson and almost Jonsson sets, we define a particular subclass of Jonsson theories. This subclass is known as the class of $cl$-normal Jonsson theories.

Definition 20.

Let τ be some given Jonsson pregeometry. A Jonsson theory T is called normal in pregeometry τ ($cl$-normal), if, for each almost Jonsson subset $X\subseteq C_T$, $C_{Fr\left(X\right)}$ is an existentially closed submodel of $C_T$.

The introduction of this definition is necessary because, when studying arbitrary Jonsson theories, there is a risk that the fragments of almost Jonsson subsets in the semantic model may not be closed under cosemanticness. This lack of closure significantly complicates the analysis. By refining the class of theories under consideration, we can obtain more accurate and comprehensive results. Moreover, this refinement still allows for a sufficient number of examples. The following theorem and corollary also provide further examples of $cl$-normal Jonsson theories.

Theorem 11.

Let T be an inductive theory complete for $\forall \exists$-sentences. Then, T is a $cl$-normal Jonsson theory.

Proof. 

Let T be an $\forall \exists$-complete Jonsson theory. Then, $T=T{h}_{\forall \exists }\left(A\right)$ for any $A\in Mod\left(T\right)$, which means that $T{h}_{\forall \exists }\left(A\right)$ is a Jonsson theory for all $A\in Mod\left(T\right)$. Therefore, T is $cl$-normal. □

Corollary 3.

For any perfect Jonsson theory T, $T^{*}$ is a $cl$-normal Jonsson theory.

Examples of theories satisfying Corollary 3 are as follows:

Example 3.

(1) 

The theory of divisible abelian groups;

(2) 

The theory of algebraically closed fields of the fixed characteristic p;

(3) 

The theory of differentially closed fields of the fixed characteristic p and others.

Thus, Section 1 covered the fundamental model-theoretic concepts and presented the facts relevant to the framework for studying Jonsson theories. Jonsson theories form a special subclass of inductive theories that exhibit certain classical model-theoretic properties with respect to the embedding of models. Key methods for studying are represented by the companion technique of A. Robinson, as well as the semantic method proposed by T. G. Mustafin. The semantic method employs the relation of cosemanticness between Jonsson theories as the primary tool for comparison and general description of theories. Additionally, specific subclasses of Jonsson theories were considered, namely the class of perfect Jonsson theories and the class of $cl$-normal Jonsson theories.

Next, we proceed to the description of the concept of the Kaiser class of a Jonsson theory, which serves as a new tool for a more precise description of cosemantic Jonsson theories. We also provide a description of perfect and $cl$-normal Jonsson theories in terms of the Kaiser class.

3. The Kaiser Class of a Jonsson Theory

In the previous section, the concepts of Jonsson sets, almost Jonsson sets, and $cl$-normal Jonsson theories were discussed. Within this framework, closures are constructed for the sets under consideration, which serve as models of the given theory. A necessary condition in this case is that the Kaiser hull of such a model must be a Jonsson theory. The description of the class of $cl$-normal Jonsson theories presents an interesting and complex task, and thus, our goal is to provide a description of the class of such models. In this context, in this work, we introduce the following definition.

Definition 21.

Let T be a Jonsson theory. A class $K_T$ of L-structures is called a Kaiser class of T if

$$K_T=\{M\mid M\in Mod\left(T\right)\mathit{and}T{h}_{\forall \exists }\left(M\right)\mathit{is}a\mathit{Jonsson}\mathit{theory}\}.$$

Based on the given definition, we can assert that the Kaiser class is a special subclass of the class of models of the considered Jonsson theory, encapsulating the ’Jonsson essence’ of the theory. In this section, we attempt to present some of the key properties of the Kaiser class and describe some of the previously introduced concepts using this concept.

Firstly, we note the following fact:

Lemma 2.

For all models $A\in K_T$, $\left|A\right|\ge \omega$.

Proof. 

We prove this fact by contradiction. Suppose there is a model $A\in K_T$ such that A is finite. It is well known (p. 323 [7]) that “an L-structure contains at most n elements, where $n<\omega$” can be formalized by a universal L-sentence”. For example, the sentence

$$\forall x\forall y\forall x\left(\right(x\ne y)\to ((x=z)\vee (y=x)\left)\right)$$

states that the L-structure contains no more that 2 elements. Then, the theory $T{h}_{\forall \exists }\left(A\right)$ contains a L-sentence ${\phi }_n$ that declares the finite cardinality of A. Then, all models $M\in Mod\left(T{h}_{\forall \exists }\left(A\right)\right)$ are finite, so $T{h}_{\forall \exists }\left(A\right)$ has no infinite models and is not a Jonsson theory. Hence, all $A\in K_T$ is an infinite model. □

The following proposition shows that for any Jonsson theory, $K_T$ is never empty.

Proposition 3.

For any Jonsson theory T, $E_T\subseteq K_T$.

The proof follows from Theorem 5.

Considering the Kaiser class of Jonsson theories, we can present some examples of models from $K_T$ for a Jonsson theory T. Suppose $T_G$ is a theory of all groups, $M\in E_{T_G}$, that is, M is an existentially closed group in $T_G$. Then, $M\in K_T$ by Proposition 3. Another example of a model that is in $K_T$ but is not existentially closed is the following. Let $T_{AG}$ be the theory of all abelian group, and let N be a divisible abelian group. It is well known that the class of existentially closed models for $T_{AG}$ is represented exactly by divisible abelian groups, then $N\in E_{T_{AG}}$. Then, $Th\left(N\right)$ is a Jonsson theory according to Theorem 5. $N\in Mod\left(T_G\right)$; therefore, $N\in K_{T_G}$. Note that N is not existentially closed over $T_G$.

From Proposition 3, it is also clear that if T is a theory whose Kaiser class is empty, this theory is not Jonsson. There are also theories whose Kaiser class is empty. The following proposition demonstrates this case.

Proposition 4.

Let T be an inductive $\forall \exists$-complete theory that is not Jonsson. Then, $K_T=⌀$.

Proof. 

Let T be a theory complete for $\forall \exists$-sentences. Then, any two models of T do not differ by $\forall \exists$-sentences, and $T=T{h}_{\forall \exists }\left(A\right)$ for any $A\in Mod\left(T\right)$. But T is not Jonsson by the condition of the proposition, so there is no $A\in Mod\left(T\right)$ such that $T{h}_{\forall \exists }\left(A\right)$ is a Jonsson theory; therefore, $K_T=⌀$. □

The example of a theory satisfying Proposition 4 is the theory of pairs of algebraically closed fields of fixed characteristic. This theory is complete and inductive, but not Jonsson.

Since the Kaiser class of a Jonsson theory is a special subclass of the models of the theory, the question of its axiomatizability arises. To address this, we examine the Kaiser hull of this class. We also describe some of its model-theoretic properties using the semantic method. This leads to the following first result:

Proposition 5.

Let T be a Jonsson theory, then $T{h}_{\forall \exists }\left(K_T\right)=K_T^0$ is a Jonsson theory cosemantic to T.

Proof. 

It is clear that $T{h}_{\forall \exists }\left(K_T\right)={\bigcap }_{i\in I}T{h}_{\forall \exists }\left(A_i\right)$, $A_i\in K_T$. By virtue of the inductiveness of Jonsson theory, $T\subseteq T{h}_{\forall \exists }\left(A_i\right)$ for any $A_i\in K_T$. Consequently, $T\subseteq T{h}_{\forall \exists }\left(K_T\right)$. Note that $T{h}_{\forall \exists }\left(K_T\right)$ is an inductive theory. Then, by Theorem 8, $T{h}_{\forall \exists }\left(K_T\right)$ is a Jonsson theory cosemantic to T. □

The concept of the Kaiser class is also important in the context of extending the framework of the semantic method. In the classical interpretation of the semantic method, two Jonsson theories are related by examining the comparison of their semantic models, that is, through the relation of cosemanticness. We can refine this approach by instead considering the theories through their Kaiser classes, introducing the following binary relation for Jonsson theories:

Definition 22.

Let $T_1$ and $T_2$ be Jonsson L-theories. $T_1$ and $T_2$ are called $K_T$-equivalent ($T_1\stackrel{\sim }{\bowtie }{T}_2$), if $K_{T_1}=K_{T_2}$.

Similar to the cosemanticness of Jonsson theories, it is clear that $K_T$-equivalence is an equivalence relation.

Through the following lemma, we establish a connection between cosemanticness and equivalence of Jonsson theories.

Lemma 3.

If Jonsson L-theories $T_1$ and $T_2$ are K-equivalent, then $T_1$ and $T_2$ are cosemantic.

Proof. 

Let $K_{T_1}=K_{T_2}$. Then, due to Theorem 4 and Proposition 3, $C_{T_1}\vDash T_2$ and $C_{T_2}\vDash T_1$. We obtain by Theorem 7 that $T_1$ and $T_2$ are cosemantic. □

The converse is generally not true. The example supporting this is provided by two theories:

Example 4.

The theory of abelian groups $T_{AG}$ and the theory $T_{AG}^{*}$ of divisible abelian groups. The class of models of $T_{AG}^{*}$ is represented by existentially closed abelian groups; however, there exist abelian groups that is not e.c. whose Kaiser hull is a Jonsson theory.

Thus, when introducing the relation of $K_T$-equivalence on the considered class of Jonsson L-theories, we obtain an additional partition within each cosemanticness class.

The next statement shows the connection between the Kaiser classes of two Jonsson theories regarding the relations of equality and cosemanticness of classes of structures.

Theorem 12.

For any Jonsson theory T, the following two conditions are equivalent:

(1) 

$T_1\stackrel{\sim }{\bowtie }{T}_2$;

(2) 

$K_{T_1}\bowtie K_{T_2}$.

Proof. 

The implication (1) → (2) is obvious. We show the implication (2) → (1). Let $K_{T_1}\bowtie K_{T_2}$, i.e., $JSp\left(K_{T_1}\right)=JSp\left(K_{T_2}\right)$. Since, by Proposition 5, $T{h}_{\forall \exists }\left(K_T\right)$ is a Jonsson theory for any Jonsson theory T, $T{h}_{\forall \exists }\left(K_{T_1}\right)\in JSp\left(K_{T_1}\right)$ and $T{h}_{\forall \exists }\left(K_{T_2}\right)\in JSp\left(K_{T_2}\right)$. Consequently, $T{h}_{\forall \exists }\left(K_{T_1}\right)=T{h}_{\forall \exists }\left(K_{T_2}\right)$. It follows that $K_{T_1}\subseteq Mod\left(T_2\right)$ and $K_{T_2}\subseteq Mod\left(T_1\right)$. Then, by virtue of Definition 21, $K_{T_1}\subseteq K_{T_2}$ and $K_{T_2}\subseteq K_{T_2}$; hence, $K_{T_1}=K_{T_2}$. So, the implication (2) → (1) is true. □

Based on the proposed facts, there arises a need to provide an example of different $K_T$-equivalent theories. Indeed, the relation of $K_T$-equivalence is nontrivial, and we demonstrate this in the following section. For this, we need the following theorem, which is a special case of Theorem 8:

Theorem 13.

Let T be a Jonsson theory, $C_T$ be the semantic model of T, $T^{\prime }$ be an inductive theory such that $T\subseteq T^{\prime }$, $K_T\subseteq Mod\left(T^{\prime }\right)$. Then, $T^{\prime }$ is a Jonsson theory which is $K_T$-equivalent to T.

Proof. 

As $C_T\in K_T$ by virtue of Theorem 3, $C_T\in Mod\left(T^{\prime }\right)$, which means by Theorem 8 $T^{\prime }$ is a Jonsson theory cosemantic to T. From $K_T\subseteq Mod\left(T^{\prime }\right)$, it follows that $K_T\subseteq K_{T^{\prime }}$. It is also clear that $Mod\left(T^{\prime }\right)\subseteq Mod\left(T\right)$; hence, $K_{T^{\prime }}\subseteq K_T$, and therefore, $K_{T^{\prime }}=K_T$, and T and $T^{\prime }$ are $K_T$-equivalent. □

From Theorem 13, we obtain two clear corollaries:

Corollary 4.

For any Jonsson theory T, T and $K_T^0$ are $K_T$-equivalent.

Corollary 5.

For any theory $T^{\prime }$ such that $T\subseteq T^{\prime }\subseteq K_T^0$, $T^{\prime }\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$.

And the following corollary allows us to give examples of different Jonsson theories that are $K_T$-equivalent.

Corollary 6.

Let T be a Jonsson L-theory and $T_{\infty }$ be an L-theory such that

$$Mod\left(T_{\infty }\right)=\{A\mid A\in Mod\left(T\right)and\left|A\right|\ge \omega \}.$$

Then $T_{\infty }$ is a Jonsson theory that is $K_T$-equivalent to T.

Proof. 

According to Theorem 8, $T_{\infty }$ is a Jonsson theory, and then $T_{\infty }$ is inductive. It is clear that $T\subseteq T_{\infty }$, and due to Lemma 2, $K_T\subseteq Mod\left(T_{\infty }\right)$; then, T and $T_{\infty }$ are $K_T$-equivalent. □

In this manner, the examples of Jonsson theories that are $K_T$-equivalent are as follows:

Example 5.

(1) 

The theory of all (abelian) groups and the theory of all infinite (abelian) groups;

(2) 

The theory of all fields of characteristic $p\ge 0$ and the theory of all infinite fields of the same characteristic;

(3) 

The theory of all differential fields of characteristic $p>0$ and the theory of all infinite differential fields of the same characteristic.

Now, we can consider the class of all theories $K_T$-equivalent to a given theory T. Here and throughout, we denote this class by ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$.

Theorem 14.

Let T be a Jonsson theory. Then, for any theory $T^{\prime }\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$, $T^{\prime }\subseteq K_T^0$.

In other words, in the class of $K_T$-equivalent theories, $K_T^0$ is the greatest theory regarding set-theoretic inclusion of theories.

Proof. 

We prove this theorem by contradiction. Let $T_1$ be a theory such that $T_1\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ and $K_T^0\subseteq T_1$. Then, there is a universal–existential L-sentence $\phi$ such that $T_1⊢\phi$ and $K_T^0⊬\phi$. Since $K_T\subseteq Mod\left(T_1\right)$, for any $A\in K_T$, it is true that $A\vDash \phi$. Consequently, $\phi \in T{h}_{\forall \exists }\left(K_T\right)$ and $K_T^0⊢\phi$, which is a contradiction. □

As previously mentioned, in this section, we address the issue of the axiomatizability of the Kaiser class. Speaking of the axiomatizability of a class K of L-structures in general, one of the central problems in Model Theory is the following question: under which conditions does $Mod\left(Th\right(K\left)\right)=K$? The following theorem represents some additional facts regarding this issue.

Theorem 15.

Let T be a Jonsson theory complete for $\forall \exists$-sentences. Then, $Mod\left(T\right)=K_T$.

Proof. 

Let $A\in Mod\left(T\right)$, then, due to $\forall \exists$-completeness of T, $T=T{h}_{\forall \exists }\left(A\right)$, which is a Jonsson theory. By virtue of the arbitrariness of $A\in Mod\left(T\right)$, it is true for any model of T; therefore, $Mod\left(T\right)=K_T$. □

As we can see, the problem of the axiomatizability of the Kaiser class naturally aligns with the classical problem of the axiomatizability of the class of existentially closed models of the given theory. This raises the question: under which conditions does $Mod\left(K_T^0\right)=K_T$? A second question arising from the theory of companions is the following: is $K_T^0$ a companion of theory T? The answer to this is provided by the following remark:

Remark 1.

$K_T^0$ is not a companion of T.

Proof. 

The proof follows from the fact that Condition 1 in Definition 9 fails for $K_T^0$. Indeed, there are examples of Jonsson theories $T_1$ and $T_2$ that are mutually model consistent and, consequently, by Theorem 1, are cosemantic, but not $K_T$-equivalent. It means that $K_{T_1}^0\ne K_{T_2}^0$. Such examples are described in Example 4. □

The following theorems provide insight into the axiomatizability of the Kaiser class.

Theorem 16.

For any Jonsson theory T, the class $K_T$ is closed under the joint embedding and amalgamation properties.

Proof. 

Firstly, let us show the closedness of $K_T$ with respect to JEP. Let $A,B\in K_T$. According to Theorem 4 and Proposition 3, $C_T\in K_T$. By the definition of a semantic model, there are isomorphic embeddings $f:A\to C_T$ and $g:B\to C_T$; therefore, $K_T$ is closed with respect to JEP.

Now, consider $K_T$ with respect to AP. Let $A,B,C\in K_T$ and $f_1:A\to B$ and $f_2:A\to C$ be isomorphic embeddings. Since T is a Jonsson theory and admits an amalgamation property, there is a model $D\in Mod\left(T\right)$ and isomorphic embeddings $g_1:B\to D$ and $g_2:C\to D$ such that $f_1·f_2=g_1·g_2$. If $D\in K_T$, then the proof is completed. If $D\notin K_T$, then we consider an isomorphic embedding $h:D\to C_T$ that exists by virtue of ${\omega }^{+}$-universality of $C_T$. Then, $g_1·h:B\to C_T$ and $g_1·h:C\to C_T$. Due to ${\omega }^{+}$-homogeneity of $C_T$, $f_1·g_1·h=f_2·g_2·h$, which means that $K_T$ is closed with respect to AP. □

Theorem 17.

Let T be a Jonsson theory. If $K_T$ is closed under the unions of chains, then $Mod\left(K_T^0\right)=K_T$.

Proof. 

By Lemma 2, $K_T$ has infinite models. According to Theorem 16, this class is also closed under AP and JEP. Then, if $K_T$ is closed under the unions of chains, there is a Jonsson theory $T^{\prime }$ such that $Mod\left(T^{\prime }\right)=K_T$. It is clear that $K_T=Mod\left(T^{\prime }\right)\subseteq Mod\left(K_T^0\right)$, then $K_T^0\subseteq T^{\prime }$. By virtue of Theorem 14, $T^{\prime }\subseteq K_T^0$; therefore, $K_T^0=T^{\prime }$. □

Next, we turn to the description of the class of perfect and the class of $cl$-normal Jonsson theories within the framework of the Kaiser class. The application of this technique to the study of perfect Jonsson theories results in a new interpretation of the perfectness criterion (Theorem 6).

Theorem 18.

Let T be a Jonsson theory. Then, the following conditions are equivalent:

(1) 

T is a perfect Jonsson theory;

(2) 

$E_T=K_{T^{*}}$.

Proof. 

According to Theorem 6, $E_T=Mod\left(T^{*}\right)$. From this, it also follows that $E_T=E_{T^{*}}$. By Proposition 3, $E_{T^{*}}\subseteq K_{T^{*}}$. Therefore, $E_{T^{*}}=K_{T^{*}}=E_T$. □

To present the results concerning $cl$-normal Jonsson theories using the Kaiser class technique, we require the following lemma, which serves as a criterion for the $cl$-normality of a Jonsson theory in the language of cosemanticness of theories:

Lemma 4.

For any Jonsson theory T, the following conditions are equivalent:

(1) 

T is a $cl$-normal Jonsson theory;

(2) 

For any almost Jonsson set $X\subseteq C_T$ of T, $Fr\left(X\right)\bowtie T$.

Proof. 

Firstly, we show the implication (1) → (2). Let T be a $cl$-normal Jonsson theory, then for any almost Jonsson set $X\subseteq C_T$, $C_{Fr\left(X\right)}$ is a model of T and an existentially closed submodel of $C_T$. By Theorem 4, $C_T$ is an existentially closed model of T. It is also clear that $C_{Fr\left(X\right)}$ is a model of T. Then, by Theorem 1, $C_{Fr\left(X\right)}$ is an existentially closed model of T. Due to Theorem 2, $C_{Fr\left(X\right)}$ and $C_T$ are elementary equivalent by $\forall \exists$-sentences, and then $C_T$ is a model of $Fr\left(X\right)$. Therefore, by Theorem 7, $T\bowtie Fr\left(X\right)$.

Now, we show the implication (2) → (1). Let T be a Jonsson theory, $X\subseteq C_T$ be an almost Jonsson set of T, and $Fr\left(X\right)\bowtie T$. Then, $C_{Fr\left(X\right)}=C_T$. It is obvious that $C_{Fr\left(X\right)}$ is an existentially closed submodel of $C_T$ and $C_{Fr\left(X\right)}\vDash T$. Therefore, T is a $cl$-normal Jonsson theory. □

The following theorem refines Theorem 8 for the case of $cl$-normal Jonsson theory and allows us to conclude that the $cl$-normality of a Jonsson theory is preserved under its Jonsson extensions.

Theorem 19.

Let T be a $cl$-normal Jonsson L-theory and $T^{\prime }$ be an inductive L-theory such that $T\subseteq T^{\prime }$ and $C_T\vDash T^{\prime }$. Then, $T^{\prime }$ is a $cl$-normal Jonsson theory cosemantic to T.

Proof. 

According to Theorem 8, $T^{\prime }$ is a Jonsson theory cosemantic to T. Let $X\subseteq C_{T^{\prime }}$ be an almost Jonsson set of theory $T^{\prime }$, $M=cl\left(X\right)$, then $M\in Mod\left(T^{\prime }\right)$, and $T{h}_{\forall \exists }\left(M\right)$ is a Jonsson theory. Since $T\subseteq T^{\prime }$, $Mod\left(T^{\prime }\right)\subseteq Mod\left(T\right)$ and $M\in Mod\left(T\right)$. Due to the cosemanticness of T and $T^{\prime }$, $X\subseteq C_T$; therefore, X is an almost Jonsson subset of the theory T. If T is a $cl$-normal Jonsson theory, then the theory $T{h}_{\forall \exists }\left(M\right)$ is cosemantic to T and $T^{\prime }$. Thus, $T^{\prime }$ is a $cl$-normal Jonsson theory. □

From Theorem 19, we have two useful corollaries:

Corollary 7.

Let T be a $cl$-normal Jonsson theory. Then, for any almost Jonsson set $X\subseteq C_T$ of T, $Fr\left(X\right)$ is a $cl$-normal Jonsson theory.

Proof. 

Let $X\subseteq C_T$ be an almost Jonsson set of T, $cl\left(X\right)=M$, then $M\in Mod\left(T\right)$, $T{h}_{\forall \exists }\left(M\right)=Fr\left(X\right)$ is a Jonsson theory. It is clear that $T\subseteq Fr\left(X\right)$ and $Fr\left(X\right)$ is an inductive theory. T is a $cl$-normal Jonsson theory, therefore $C_{Fr\left(X\right)}$ is an existentially closed submodel of $C_T$. By Lemma 4, $Fr\left(X\right)\bowtie T$. Then, by Theorem 19, $Fr\left(X\right)$ is a $cl$-normal Jonsson theory. □

Corollary 8.

Let T be a Jonsson theory, $T_1,T_2\in {\left[T\right]}_{\bowtie }$ be $cl$-normal Jonsson theories, and $T_3$ be a theory such that $T_1\subseteq T_3\subseteq T_2$. Then, $T_3$ is a $cl$-normal Jonsson theory cosemantic to T.

Proof. 

Since $T_1$ and $T_2$ are inductive theories, $T_2$ is also an inductive theory. By the condition of the theorem, $C_{T_1}=C_{T_2}=C_T$. Then, $C_T\vDash T_3$. By Theorem 19, it follows that $T_3$ is a $cl$-normal Jonsson theory cosemantic to T. □

It is well known that the cosemanticness class of a perfect Jonsson theory contains only perfect Jonsson theories, that is, it is closed under the property of perfectness. This fact directly follows from the definitions of a perfect Jonsson theory and cosemanticness of Jonsson theories. A similar fact can be obtained for the $K_T$-equivalence class of the Jonsson theory T with respect to $cl$-normality:

Theorem 20.

Let T be a Jonsson theory and ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ be the $K_T$-equivalence class of T. Then, if T is a $cl$-normal Jonsson theory, any theory $T^{\prime }\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ is a $cl$-normal Jonsson theory.

Proof. 

Let T be a $cl$-normal Jonsson theory, $T^{\prime }\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$. Due to Lemma 3, $T^{\prime }$ is cosemantic to T. Let $X\subseteq C_{T^{\prime }}$ be an almost Jonsson set of $T^{\prime }$, $cl\left(X\right)=M$, then $M\in Mod\left(T^{\prime }\right)$ and $T{h}_{\forall \exists }\left(M\right)$ is a Jonsson theory. It follows that $M\in K_{T^{\prime }}=K_T$; therefore, $M\in Mod\left(T\right)$. Then, X is an almost Jonsson set of T. By virtue of $cl$-normality of T, $T{h}_{\forall \exists }\left(M\right)$ is cosemantic to T and $T^{\prime }$. Therefore, $T^{\prime }$ is a $cl$-normal Jonsson theory. Due to the arbitrariness of $T^{\prime }$, all theories in ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ are $cl$-normal Jonsson theories. □

Finally, we present the criterion of $cl$-normality for perfect Jonsson theories in terms of the Kaiser class.

Theorem 21.

Let T be a perfect Jonsson theory. Then, the following conditions are equivalent:

(1) 

T is a $cl$-normal Jonsson theory;

(2) 

For any almost Jonsson set $X_i\subseteq C_T$, $i\in I$, ${\bigcap }_{i\in I}{K}_{Fr\left(X_i\right)}\ne ⌀$.

Proof. 

Firstly, we show (1) → (2). Let T be a $cl$-normal perfect Jonsson theory. Then, by Lemma 4, for any almost Jonsson set $X\subseteq C_T$ of T, $Fr\left(X\right)\bowtie T$; therefore, $C_T\vDash Fr\left(X\right)$. According to Theorem 4 and Proposition 3, $C_T\in K_T$ and $C_T\in K_{Fr\left(X\right)}$; therefore, $K_T\cap K_{Fr\left(X\right)}\ne ⌀$. Since $C_T\vDash Fr\left(X_i\right)$ for all $i\in I$, $C_T\in {\bigcap }_{i\in I}{K}_{Fr\left(X_i\right)}$, so this intersection is nonempty.

Now, let $X_i\subseteq C_T$, $i\in I$ be all almost Jonsson sets of T and let ${\bigcap }_{i\in I}{K}_{Fr\left(X_i\right)}\ne ⌀$. Let X be a domain of $C_T$. This set is ∃-definable, $cl\left(X\right)=C_T$ and $Fr\left(X\right)=T{h}_{\forall \exists }\left(C_T\right)$ is a Jonsson theory by Theorems 2 and 5. Therefore, X is an almost Jonsson set of T (and even a Jonsson set, since $C_T$ is an existentially closed model of T). By the condition of the theorem, T is a perfect Jonsson theory, then by Theorem 6, $Fr\left(X\right)=T{h}_{\forall \exists }\left(C_T\right)=T^{*}$ and $Mod\left(Fr\left(X\right)\right)=E_T$, then $K_{Fr\left(X\right)}=E_T$. Since ${\bigcap }_{i\in I}{K}_{Fr\left(X_i\right)}\ne ⌀$, $C_T\vDash Fr\left(X_i\right)$ for all $i\in I$. It is also clear that for any $i\in I$, $T\subseteq Fr\left(X_i\right)$ and $Fr\left(X_i\right)$ are inductive theories. Then, by Theorem 8, every fragment $Fr\left(X_i\right)$ is a Jonsson theory cosemantic to T. Hence, T is a $cl$-normal Jonsson theory by Lemma 4. □

To conclude, in this section, the concept of the Kaiser class of a Jonsson theory was introduced, along with a description of its basic properties. Additionally, the notions of perfectness and $cl$-normality of a Jonsson theory were presented using this concept.

In the next section, we use the obtained results to construct algebraic structures within the class of cosemanticness of an arbitrary fixed Jonsson theory.

4. The Lattices of Cosemantic Jonsson Theories

This section pertains to the question of the algebraization of first-order theories. We consider theories as elements of certain sets, on which we define a lattice algebraic structure. It is known that first-order theories form a lattice under the operations of union and intersection, provided that the theories being considered are consistent in their union. We focus on a specific subclass of theories where the lattice structure is naturally defined and admits some well-described algebraic properties. This approach will later enable us to apply the appropriate methods of Universal Algebra to study the classes of theories we consider.

Firstly, let us recall some basic definitions related to lattice theory [15].

A lattice L is a partially ordered set where any two elements $a,b\in L$ have both a least upper bound (join) $a\vee b$ and a greatest lower bound (meet) $a\wedge b$. These operations satisfy the following properties:

(1)

Idempotence: $a\vee a=a\mathrm{and}a\wedge a=a$.

(2)

Commutativity: $a\vee b=b\vee a\mathrm{and}a\wedge b=b\wedge a$.

(3)

Associativity: $a\vee (b\vee c)=(a\vee b)\vee c\mathrm{and}a\wedge (b\wedge c)=(a\wedge b)\wedge c$.

(4)

Absorption: $a\vee (a\wedge b)=a\mathrm{and}a\wedge (a\vee b)=a$.

A lattice L is called distributive if the following two conditions hold for all $a,b,c\in L$:

(1)

Distributivity of join over meet: $a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)$.

(2)

Distributivity of meet over join: $a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$.

A lattice L is said to be convex if for every $a,b,c\in L$, with $a\le c\le b$, the following condition holds:

$$a\vee b=c\mathrm{implies}a\le c\le b.$$

In this section, we fix an arbitrary Jonsson theory T and consider its cosemantic class. Here and henceforth, we denote by ${\left[T\right]}_{\bowtie }$ the class of all L-theories that are cosemantic with T, and by ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ the class of all theories of the given language that are $K_T$-equivalent to T.

Let us introduce the operations “∨” and “∧” for theories as follows. Let T and $T^{\prime }$ be L-theories. Let

$$T\wedge T^{\prime }=\{\phi \wedge {\phi }^{\prime }|\phi \in T,{\phi }^{\prime }\in T^{\prime }\},$$

if this theory is consistent. This theory is logically equivalent to $T\cup T^{\prime }$, and the class of models of this theory consists of L-structures that are models of T and $T^{\prime }$ simultaneously. Let us show that. Let M be a model of $T_1\wedge T_2$, let $\phi$ be any sentence from T, and let $\psi$ be any sentence from $T^{\prime }$. Then, $M\vDash \phi \wedge \psi$; therefore, $M\vDash \phi$ and $M\vDash \psi$, which means that $M\vDash T$ and $M\vDash T^{\prime }$. Hence, $M\vDash T\cup T^{\prime }$. We obtain that $Mod(T\wedge T^{\prime })\subseteq Mod(T\cup T^{\prime })$. Now, let N be a model of $T\cup T^{\prime }$. It follows that for any sentence $\phi \in T$ and any sentence $\psi \in T^{\prime }$, $N\vDash \phi$ and $N\vDash \psi$; therefore $N\vDash \phi \wedge \psi$, and $N\vDash T\wedge T^{\prime }$. Thus, $Mod(T\cup T^{\prime })\subseteq Mod(T\wedge T^{\prime })$, and $Mod(T\cup T^{\prime })=Mod(T\wedge T^{\prime })$, and $T\wedge T^{\prime }$ is logically equivalent to $T\cup T^{\prime }$. From this, we see that $Mod(T\wedge T^{\prime })$ is exactly L-structures that satisfy both T and $T^{\prime }$, that is $Mod(T\wedge T^{\prime })=Mod\left(T\right)\cap Mod\left(T^{\prime }\right)$.

Let

$$T\vee T^{\prime }=\{\phi \vee {\phi }^{\prime }|\phi \in T,{\phi }^{\prime }\in T^{\prime }\}.$$

The class of models of $T\vee T^{\prime }$ is represented by L-structures that are models of T or $T^{\prime }$. This fact follows from Lemma 2 of [13]. Note that $T\vee T^{\prime }$ is logically equivalent to $T\cap T^{\prime }$. Let us show this. It is clear that $Mod\left(T\right)\cup Mod\left(T^{\prime }\right)\subseteq Mod(T\cap T^{\prime })$. Let M be a model of $T\cap T^{\prime }$, which means that $M\vDash \psi$ for any $\psi \in T\cap T^{\prime }$. Suppose $M\notin Mod\left(T\right)\cup Mod\left(T^{\prime }\right)$, then $M\notin Mod\left(T\right)$ and $M\notin Mod\left(T^{\prime }\right)$; therefore, there are L-sentences $\alpha \in T$ and $\beta \in T^{\prime }$ such that $M⊭\alpha$ and $M⊭\beta$. Then, $M\vDash \neg \alpha$ and $M\vDash \neg \beta$, and we have that $M\vDash \neg (\alpha \vee \beta )$. But $\alpha \in T$, and therefore, $T⊢\alpha \vee \beta$, which means that $\alpha \vee \beta \in T$, since T is deductively closed set of sentences. The same is for $T^{\prime }$: $\beta \in T^{\prime }$ and $\alpha \vee \beta \in T^{\prime }$, then $\alpha \vee \beta \in T\cap T^{\prime }$ and $M\vDash \alpha \vee \beta$, which is a contradiction. Hence, $M\in Mod\left(T\right)\cup Mod\left(T^{\prime }\right)$ and $Mod(T\cap T^{\prime })\subseteq Mod\left(T\right)\cup Mod\left(T^{\prime }\right)$; therefore, $Mod(T\cap T^{\prime })=Mod\left(T\right)\cup Mod\left(T^{\prime }\right)$.

The foundational result from which we begin is the following theorem proved by Ye T. Mustafin:

Theorem 22

([10]). Let T and $T^{\prime }$ be cosemantic Jonsson theories. Then, $T\vee T^{\prime }$ is also a Jonsson theory that is cosemantic to T and $T^{\prime }$.

Another key result upon which we build was established by the first and second authors in [6]:

Theorem 23.

Let K be a class of L-structures and $JSp\left(K\right)$ be the Jonsson spectrum of K. Then, any cosemanticness class $\left[T\right]\in JSp{\left(K\right)}_{/\bowtie }$ is a lattice with respect to operations “∨” and “∧”.

Let us illustrate the action of these operations for some sample cosemantic Jonsson theories.

Let $T_G$ be the theory of all groups with the binary operation “+”, and let $\phi$ be the following L-sentence:

$$\exists x\exists y(x+y\ne y+x),$$

which declares that a group is not abelian, and let $\psi$ be the following sentence:

$$\exists x\exists y\exists z\left(\right(x\ne y)\wedge (y\ne z)\wedge (x\ne z)\wedge (y\ne z\left)\right),$$

which states that a group contains more than 2 elements. Then, $T_G\cup \left\{\phi \right\}$ and $T_G\cup \left\{\psi \right\}$ are Jonsson theories that are cosemantic to $T_G$ according to Theorem 9. Then,

$$(T_G\cup \left\{\phi \right\})\wedge (T_G\cup \left\{\psi \right\})$$

is logically equivalent to $T_G\cup \left\{\phi \right\}\cup \left\{\psi \right\}$, which is also a theory cosemantic to $T_G$.

As for operation “∨”, the class of models of $T_G\cup \left\{\phi \right\}\vee T_G\cup \left\{\psi \right\}$ is the class of all L-structures such that each of them is a nonabelian group or one that contains more than 2 elements; therefore, $T_G\cup \left\{\phi \right\}\vee T_G\cup \left\{\psi \right\}$ is logically equivalent to $T_G\cup \{\phi \vee \psi \}$, which is also a Jonsson theory cosemantic to $T_G$, since $\{\phi \vee \psi \}$ is equivalent to existential sentence.

Let ${\left[T\right]}_{\bowtie }$ be a class of all L-theories cosemantic to T, then Theorem 23 also holds, and ${\left[T\right]}_{\bowtie },\wedge ,\vee$ is a lattice. We denote this lattice by $L^{\bowtie }$. The following fact is easy to see:

Proposition 6.

$L^{\bowtie }$ is a distributive lattice.

The proof follows from the fact that $T_1\wedge T_2=T_1\cup T_2$ and $T_1\vee T_2=T_1\cap T_2$ for any $T_1,T_2\in {\left[T\right]}_{\bowtie }$, where ∪ and ∩ are set-theoretic operations of union and intersection, respectively.

Now, let us consider the $K_T$-equivalence class of the theory T. We show that the lattice structure can be introduced in this class, too.

Theorem 24.

For any Jonsson theory T, ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ is a convex distributive lattice with respect to operations ∧ and ∨.

Proof. 

Firstly, we show the closedness of ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ with respect to operation ∧. Let $T_1,T_2\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$. It is obvious that $T_1\wedge T_2=T_1\cup T_2$ is an $\forall \exists$-axiomazited (and consequently inductive) theory. $K_T\subseteq Mod\left(T_1\right)$ and $K_T\subseteq Mod\left(T_2\right)$, which means that $K_T\subseteq Mod(T_1\wedge T_2)$. It is also clear that $T_1,T_2\subseteq T_1\wedge T_2$. Hence, by Theorem 13, $T_1\wedge T_2$ is a Jonsson theory that is $K_T$-equivalent to $T_1$ and $T_2$, and $T_1\wedge T_2\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$.

Now, let us consider ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ with respect to operation ∨. Let $T_1,T_2\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$ again. It follows that $T_1$ and $T_2$ are cosemantic Jonsson theories by 3. According to Theorem 22, $T_1\vee T_2$ is a Jonsson theory cosemantic to $T_1$ and $T_2$, that is, $T_1\vee T_2\in {\left[T\right]}_{\bowtie }$. It is also clear that $Mod(T_1\vee T_2)=Mod\left(T_1\right)\cup Mod\left(T_2\right)$. Since $K_{T_1}=K_{T_2}=K_T$, there is no other models $A\in Mod\left(T\right)$ in $Mod\left(T_1\right)$ and $Mod\left(T_2\right)$ such that $A\notin K_T$ and $T{h}_{\forall \exists }\left(A\right)$ is a Jonsson theory. Therefore, $K_{T_1\vee T_2}=K_T$ and $T_1\vee T_2\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$.

We assume $sup(T_1,T_2)=T_1\wedge T_2$ and $inf(T_1,T_2)=T_1\vee T_2$.

The distributivity is obvious, since $T_1\wedge T_2=T_1\cup T_2$ and $T_1\vee T_2=T_1\cap T_2$, where ∪ and ∩ are set-theoretic operations of union and intersection, respectively.

Let us show the convexity of ${\left[T\right]}_{\stackrel{\sim }{\bowtie }}$. Let $T_1,T_2\in ({\left[T\right]}_{\stackrel{\sim }{\bowtie }},\wedge ,\vee )$, $T_3$ be an L-theory such that $T_1\subseteq T_3\subseteq T_2$. Then, $T_3$ is an inductive theory. It is also clear that $Mod\left(T_2\right)\subseteq Mod\left(T_3\right)\subseteq Mod\left(T_1\right)$, $K_{T_2}\subseteq K_{T_3}\subseteq K_{T_1}$. $K_{T_1}=K_{T_2}=K_T$, consequently $K_{T_3}=K_T$, then by Theorem 13, $T_3\stackrel{\sim }{\bowtie }$ and $T_3\in {\left[T\right]}_{\stackrel{\sim }{\bowtie }}$.

Thus, $({\left[T\right]}_{\stackrel{\sim }{\bowtie }},\wedge ,\vee )$ is a convex distributive lattice. □

Applying Corollary 5 to Theorem 24, we obtain the following remark:

Remark 2.

$({\left[T\right]}_{\stackrel{\sim }{\bowtie }},\wedge ,\vee )$ is a convex sublattice of $({\left[T\right]}_{\bowtie },\wedge ,\vee )$.

Further, we denote the lattice $({\left[T\right]}_{\stackrel{\sim }{\bowtie }},\wedge ,\vee )$ as $L_{\stackrel{\sim }{\bowtie }}$.

Continuing the study of the possibility of introducing a lattice structure on the subclasses of the cosemanticness class of T, we obtain the following result concerning the $cl$-normal Jonsson theories within this class.

Theorem 25.

Let T be a Jonsson theory, ${\left[T\right]}_{\bowtie }$ be the cosemanticness class of T, ${\left[T\right]}_{\bowtie }^N\subseteq {\left[T\right]}_{\bowtie }$ be a class of all $cl$-normal Jonsson theories of ${\left[T\right]}_{\bowtie }$, if it is nonempty. Then, ${\left[T\right]}_{\bowtie }^N$ is a distributive lattice with respect to operations ∧ and ∨.

Proof. 

Let $T_1,T_2\in {\left[T\right]}_{\bowtie }^N$. By Theorem 23, $T_1\wedge T_2\in {\left[T\right]}_{\bowtie }$, that is, $C_{T_1}=C_{T_2}=C_T$. It is obvious that $T_1\wedge T_2$ is an inductive theory, $T_1\subseteq T_1\wedge T_2$ and $T_2\subseteq T_1\wedge T_2$; therefore, by Theorem 19, $T_1\wedge T_2$ is a $cl$-normal Jonsson theory, and ${\left[T\right]}_{\bowtie }^N$ is closed under the operation ∧.

Now, we show that ${\left[T\right]}_{\bowtie }^N$ is closed under ∨ as well. In Theorem 23, it is shown that $T_1\vee T_2\in {\left[T\right]}_{\bowtie }$, so $C_T=C_{T_1\vee T_2}$. It is also clear that $Mod(T_1\vee T_2)=Mod\left(T_1\right)\cup Mod\left(T_2\right)$. Let $X\subseteq C_{T_1\vee T_2}$ be an almost Jonsson set of $T_1\vee T_2$, $M=cl\left(X\right)$, then $M\in Mod(T_1\vee T_2)$ and $Fr\left(X\right)=T{h}_{\forall \exists }$ is a Jonsson theory. Then, $M\in Mod\left(T_1\right)\cup Mod\left(T_2\right)$; therefore, $M\in Mod\left(T_1\right)$ or $M\in Mod\left(T_2\right)$. Let $M\in Mod\left(T_1\right)$, then we may consider X as an almost Jonsson set of $T_1$ and $T_1\vee T_2$. $T_1$ is a $cl$-normal Jonsson theory, hence $Fr\left(X\right)\bowtie T_1\bowtie T_1\vee T_2$. It follows that $T_1\vee T_2$ is a $cl$-normal Jonsson theory. It is the same when $M\in Mod\left(T_2\right)$.

We assume $sup(T_1,T_2)=T_1\wedge T_2$ and $inf(T_1,T_2)=T_1\vee T_2$.

The distributivity of the obtained lattice is clear, since $T_1\wedge T_2=T_1\cup T_2$ and $T_1\vee T_2=T_1\cap T_2$, where ∪ and ∩ are set-theoretic operations of union and intersection, correspondingly.

Thus, $({\left[T\right]}_{\bowtie }^N,\wedge ,\vee )$ is a distributive lattice. □

We apply Corollary 8 to Theorem 25 and obtain the following fact:

Corollary 9.

$({\left[T\right]}_{\bowtie }^N,\wedge ,\vee )$ is a convex sublattice of $({\left[T\right]}_{\bowtie },\wedge ,\vee )$.

Hereafter, we denote the lattice $({\left[T\right]}_{\bowtie }^N,\wedge ,\vee )$ as $L^{N\bowtie }$.

Thus, we examined the lattice structure of the theories of a given class of cosemanticness. By applying the semantic method in the study of Jonsson theories, we aim to generalize the results obtained, transferring properties from individual theories to equivalence classes of these theories whenever possible. This approach allows us to extend the scope of our research and, in the future, apply the results to a broader set of problems. We now apply this approach once more. Specifically, we consider the set of all $K_T$-equivalence classes of theories from the given class of cosemanticness ${\left[T\right]}_{\bowtie }$ and demonstrate that a lattice structure can also be defined on this set. To introduce a partial order on this set, we first require the following theorem:

Theorem 26.

Let T be a Jonsson theory, ${\left[T\right]}_{\bowtie }$ be the cosemanticness class of T. Then the relation of $K_T$-equivalence is congruent on ${\left[T\right]}_{\bowtie }$ with respect to operations ∧ and ∨.

Proof. 

As mentioned before, $K_T$-equivalence is an equivalence relation. We need to show that $K_T$-equivalence is closed under operations ∧ and ∨.

Let $T_1,T_1^{\prime }\in {\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}$, $T_2,T_2^{\prime }\in {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$. Consider $T_1\wedge T_2$. It is clear that $Mod(T_1\wedge T_2)=Mod\left(T_1\right)\cap Mod\left(T_2\right)$ and $K_{T_1\wedge T_2}=K_{T_1}\cap K_{T_2}$. Similarly, considering $T_1^{\prime }\wedge T_2^{\prime }$, we obtain $K_{T_1^{\prime }\wedge T_2^{\prime }}=K_{T_1^{\prime }}\cap K_{T_2^{\prime }}=K_{T_1}\cap K_{T_2}$, and consequently, $K_{T_1\wedge T_2}=K_{T_1^{\prime }\wedge T_2^{\prime }}$, i.e., the theories $T_1\wedge T_2$ and $T_1^{\prime }\wedge T_2^{\prime }$ belong to the same $K_T$-equivalence class in ${\left[T\right]}_{\bowtie }$.

Now, let us consider $T_1\vee T_2$. It is known that $Mod(T_1\vee T_2)=Mod\left(T_1\right)\cup Mod\left(T_2\right)$; therefore, $K_{T_1\vee T_2}=K_{T_1}\cup K_{T_2}$. Considering $T_1^{\prime }\vee T_2^{\prime }$, we obtain that $K_{T_1^{\prime }\vee T_2^{\prime }}=K_{T_1^{\prime }}\cup K_{T_2^{\prime }}=K_{T_1}\cup K_{T_2}$; hence, $K_{T_1\vee T_2}=K_{T_1^{\prime }\vee T_2^{\prime }}$, that is, the theories $T_1\vee T_2$ and $T_1^{\prime }\vee T_2^{\prime }$ also belong to the same $K_T$-equivalence class. □

Let T be a Jonsson theory, ${\left[T\right]}_{\bowtie }$ be the cosemanticness class of T, ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_2\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$ be $K_T$-equivalence classes in ${\left[T\right]}_{\bowtie }$. Let us introduce the binary operation for $K_T$-equivalence classes in ${\left[T\right]}_{\bowtie }$ as follows. We say that ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}={\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$, if

$$T_3=T_1\wedge T_2\in {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }},$$

where $T_1$ is any theory of ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}$, $T_2$ is any theory of ${\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$.

Similarly, we introduce the operation ∨ for $K_T$-equivalence classes ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_2\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$ on ${\left[T\right]}_{\bowtie }$ as follows: ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\vee {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}={\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$, if

$$T_3=T_1\vee T_2\in {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }},$$

where $T_1$ is any theory of ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}$, $T_2$ is any theory of ${\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$.

It is clear that the introduced operations ∧ and ∨ for $K_T$-equivalence classes ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}$ and ${\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$ are equivalent to the operations of set-theoretic intersection and union of $K_{T_1}$ and $K_{T_2}$, correspondingly,

$${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}={\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\iff K_{T_1}\cap K_{T_2}=K_{T_3},$$

and, similarly,

$${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\vee {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}={\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\iff K_{T_1}\cup K_{T_2}=K_{T_3}.$$

Consider the cosemanticness class ${\left[T\right]}_{\bowtie }$ of a Jonsson theory T. Introducing the relation of $K_T$-equivalence on ${\left[T\right]}_{\bowtie }$, we obtain a factor-set of ${\left[T\right]}_{\bowtie }$ by $K_T$-equivalence. Let ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$ belong to this factor-set. We introduce the partial order relation ≤ between the $K_T$-equivalence classes on ${\left[T\right]}_{\bowtie }$ as follows:

$${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\le {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}\iff K_{{\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}}\subseteq K_{{\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}}.$$

By $K_{{\left[T_i\right]}_{\stackrel{\sim }{\bowtie }}}$, we denote the Kaiser class of any theory $T_i\in {\left[T_i\right]}_{\stackrel{\sim }{\bowtie }}$. Thus, we may consider the obtained factor-set as a partially ordered set. Then, the following theorem holds.

Theorem 27.

The factor-set of ${\left[T\right]}_{\bowtie }$ by $K_T$-equivalence is a distributive lattice with respect to operations ∧ and ∨.

Proof. 

Let ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$ be $K_T$-equivalence classes in ${\left[T\right]}_{\bowtie }$. Then, by Theorem 26, there is a unique $K_T$-equivalence class ${\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$ in ${\left[T\right]}_{\bowtie }$ such that $T_3=T_1\wedge T_2\in {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$, that is, the factor-set of ${\left[T\right]}_{\bowtie }$ by $K_T$-equivalence is closed under the operation ∧. The same is for the operation ∨. Therefore, the given factor-set is a lattice. We denote this lattice as $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$.

Let us show the distributivity of $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$. Let $T_i\in {\left[T_i\right]}_{\stackrel{\sim }{\bowtie }}$, $i=\overline{1,5}$,

$${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge \left({\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}\vee {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\right)={\left[T_4\right]}_{\stackrel{\sim }{\bowtie }}.$$

Then $K_{T_4}=K_{T_1}\cap (K_{T_2}\cup K_{T_2})$. Now, we assume that

$$\left({\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}\right)\vee \left({\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\right)={\left[T_5\right]}_{\stackrel{\sim }{\bowtie }}.$$

Then $K_{T_5}=(K_{T_1}\cap K_{T_2})\cup (K_{T_1}\cap K_{T_3})=K_{T_1}\cap (K_{T_2}\cup K_{T_2})=K_{T_4}$. Therefore, $K_{T_4}=K_{T_5}$ and

$${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge \left({\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}\vee {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\right)=\left({\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}\right)\vee \left({\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\right),$$

so the lattice $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$ is distributive. □

As is shown in Theorem 20, each $K_T$-equivalence class in ${\left[T\right]}_{\bowtie }$ is closed under $cl$-normality of Jonsson theories. In this manner, we say that a $K_T$-equivalence class in ${\left[T\right]}_{\bowtie }$ is $cl$-normal if it contains a $cl$-normal Jonsson theory (and, consequently, all theories in this class are $cl$-normal). Let us consider the set of all $cl$-normal $K_T$-equivalence classes in ${\left[T\right]}_{\bowtie }$. It is clear that this set is a subset of $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$. Introducing the partial order relation ≤ and the operations ∧ and ∨ as it was performed before, we obtain the following theorem, which describes the algebraic structure of the set of $cl$-normal $K_T$-equivalence classes in ${\left[T\right]}_{\bowtie }$.

Theorem 28.

$cl$-normal $K_T$-equivalence classes of cosemantic Jonsson theories form a distributive lattice with respect to the operations ∧ and ∨.

Proof. 

Let ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }},{\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$ be $cl$-normal $K_T$-equivalence classes of Jonsson theories cosemantic to T. Considering ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$, we obtain, by Theorem 27, some $K_T$-equivalence class ${\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\in L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$. Let us show that ${\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$ is a $cl$-normal $K_T$-equivalence class. As it was mentioned before,

$${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}={\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}\iff K_{T_1}\cap K_{T_2}=K_{T_3}.$$

Let $T_3\in {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$, and let $X\subseteq C_{T_3}=C_T$ be an almost Jonsson subset of $T_3$, $M=cl\left(X\right)$, then $Fr\left(X\right)$ is a Jonsson theory. $M\in K_{T_3}=K_{T_1}\cap K_{T_2}$; hence, $M\in K_{T_1}$ or $M\in K_{T_2}$. If $M\in K_{T_1}$, then we can consider $X\subseteq C_{T_1}=C_T$ as an almost Jonsson subset of $T_1$. ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}$ is a $cl$-normal class, therefore $Fr\left(X\right)\bowtie T_1$ for all $T_1\in {\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}$, and $T_1\bowtie T_3$, which means that $Fr\left(X\right)\bowtie T_3$ for all $T_3\in {\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$. Therefore, $T_3$ is a $cl$-normal Jonsson theory, which also holds for any theory from ${\left[T_3\right]}_{\stackrel{\sim }{\bowtie }}$ due to Theorem 20. It is the same when $M\in K_{T_2}$.

The fact that ${\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\vee {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$ is a $cl$-normal class can be proved similarly. Thus, we obtained the lattice of $cl$-normal $K_T$-equivalence classes ${\left[T\right]}_{\bowtie }$ of cosemantic Jonsson theories. Let us denote it as $L_{\stackrel{\sim }{\bowtie }}^{N\bowtie }$.

The proof of distributivity of $L_{\stackrel{\sim }{\bowtie }}^{N\bowtie }$ is similar to proving the distributivity of $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$ in Theorem 27. □

Theorem 28 implies the following facts:

Corollary 10.

$L_{\stackrel{\sim }{\bowtie }}^{N\bowtie }$ is a sublattice of $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$.

Corollary 11.

$L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$ is a homomorphic image of $L^{\bowtie }$.

Proof. 

Let us consider the following map: $f:{\left[T_i\right]}_{\stackrel{\sim }{\bowtie }}\to T_i$, where $T_i$ is some fixed theory from ${\left[T_i\right]}_{\stackrel{\sim }{\bowtie }}$. Then, for any $T_1\in {\left[T_1\right]}_{\stackrel{\sim }{\bowtie }},T_2\in {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}\in L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$, $T_1\wedge T_2=T_3$, where $T_3\in {\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\wedge {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$, and $T_1\vee T_2=T_4$, where $T_4\in {\left[T_1\right]}_{\stackrel{\sim }{\bowtie }}\vee {\left[T_2\right]}_{\stackrel{\sim }{\bowtie }}$. Therefore, f is an algebraic isomorphism of two lattices. It is clear that the theories $T_i\in {\left[T_i\right]}_{\stackrel{\sim }{\bowtie }}$ that correspond to their $K_T$-equivalence classes form a sublattice in $L^{\bowtie }$. Then, $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$ is a sublattice of $L^{\bowtie }$. □

By similar way, we obtain the following corollary:

Corollary 12.

$L_{\stackrel{\sim }{\bowtie }}^{N\bowtie }$ is a homomorphic image of $L^{N\bowtie }$.

Now, let us summarize all the statements of Section 3 concerning the embeddings of the considered lattices.

Let T be a Jonsson L-theory, then $L^{\bowtie }$ is the lattice of all theories that cosemantic to T. The following cosemantic theories’ structures are sublattices of $L^{\bowtie }$:

(1)

The lattice $L_{\stackrel{\sim }{\bowtie }}$ of theories of an arbitrary $K_T$-equivalence class in $L^{\bowtie }$;

(2)

The lattice $L^{N\bowtie }$ of all $cl$-normal theories in $L^{\bowtie }$.

We also obtain that the lattice $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$ of all $K_T$-equivalence classes in $L^{\bowtie }$ is a homomorphic image of $L^{\bowtie }$, and the lattice $L_{\stackrel{\sim }{\bowtie }}^{N\bowtie }$ of all $cl$-normal $K_T$-equivalence classes in $L^{\bowtie }$ is a sublattice of $L_{\stackrel{\sim }{\bowtie }}^{\bowtie }$.

Note that the algebraic monomorphism $g:L_{\stackrel{\sim }{\bowtie }}^{N\bowtie }\to L^{N\bowtie }$ can be naturally continued to a monomorphism $g^{\prime }:L_{\stackrel{\sim }{\bowtie }}^{\bowtie }\to L^{\bowtie }$.

Finally, we would like to draw the attention of our readers to the following ideas, which we believe are relevant and interesting. We are reffering to the transfer of the main notions and results of this article, i.e., the Kaiser class and $cl$-normality of a Jonsson theory for further investigation in the framework of Positive Model Theory. In particular, one can find the start points of positive Jonsson theories in [16,17].