Almost Existentially Closed Models in Positive Logic

Abstract

This paper explores the concept of almost positively closed models in the framework of positive logic. To accomplish this, we initially define various forms of the positive amalgamation property, such as h-amalgamation and symmetric and asymmetric amalgamation properties. Subsequently, we introduce certain structures that enjoy these properties. Following this, we introduce the concepts of -almost positively closed and -weekly almost positively closed. The classes of these structures contain and exhibit properties that closely resemble those of positive existentially closed models. In order to investigate the relationship between positive almost closed and positive strong amalgamation properties, we first introduce the sets of positive algebraic formulas and and the properties of positive strong amalgamation. We then show that if a model of a theory is a -weekly almost positively closed, then is a positive strong amalgamation basis of , and if is a positive strong amalgamation basis of , then is -weekly almost positively closed.

1. Introduction

The notion of strong amalgamation base in the framework of the general model theory was defined by Bacsich in ([1, 2]). He proved that every strong amalgamation basis of a universal theory with the amalgamation property is algebraically closed in the sense of Jonsson ([3]) and Robinson ([4]). In ([5]), Eklof has shown that the converse is true in general. He established necessary conditions for members of a special class to be strong amalgamation basis, even when is not an elementary class. These conditions are expressed in terms of a strong notion of algebraically closed structures, introduced in ([5]), and utilizing the concept of closure operators.

In the conventional models theory, the strong amalgamation property is a characteristic that a structure can possess within its class of extensions. This essentially means that, for any extensions and of , there exists a common extension of and such that .

The positive model theory is concerned essentially with the study of h-inductive theories which are built without the use of the negation. Considering positive formulas and homomorphisms instead of embeddings, positive logic generates new situations beyond the scope of logic with negation.

Consequently, when examining the property of amalgamation, homomorphisms are predominantly utilized instead of embeddings. Therefore, the application of the principle of strong amalgamation mentioned at the end of the first paragraph of this introduction naturally leads to introduce the concepts of “positive strong amalgamation” and “h-strong amalgamation.”

In this paper, we will explore one of the specific aspects of positive logic which embodies the notions of algebraic closedness and strong amalgamation and undertake to study some interactions between these new notions inspired directly or indirectly from the works of Bacsich ([1]). In the first section, after summarising the necessary background of the positive model theory, we introduce the general form of symmetric and asymmetric amalgamations. We show that the model completeness of an h-inductive theory can be characterized by a form of symmetric amalgamation. The second section is devoted to the notions of almost positively closed models and a special class of positive formulas called -closed formulas. Note that the terminology “closed formula” here has different meaning of the notion of formulas without free variables. We analyse the class of almost positively closed model and present a characterization through some properties of the class of the -closed formulas. In the third section, we introduce the notions of positive strong amalgamation and h-strong amalgamation properties. We show that the class of almost positively closed has the positive strong amalgamation property. Furthermore, we give a syntactic characterization of positive strong amalgamation bases.

2. Positive Model Theory

2.1. Basic Definitions and Notations

In this subsection, we briefly introduce the basic terminology related to the positive logic. For more details, the reader is referred to ([6–8]).

Let be a first order language that contains the symbol of equality and a constant denoting the antilogy. The quantifier-free positive formulas are obtained from atomic formulas using the connectives and . The positive formulas are built from quantifier-free positive formulas using the logical operators and quantifier and , respectively. Eventually, the positive formulas are of the form , where is quantifier-free formula. The variables are said to be free. Also, a sentence is a formula without free variables.

A sentence is said to be h-inductive if it is a finite conjunction of sentences of the following form:where and are positive formulas. The h-universal sentences are the negation of positive sentences.

Let and be two -structures. A map from to is a homomorphism if for every tuple and for every atomic formula ; if , then . So, we say that is a continuation of .

An embedding of into is a homomorphism such that for every and , an atomic formula, if , then . The homomorphism is said to be an immersion whenever for every and a positive formula, if , then .

A class of -structures is said to be h-inductive if it is closed under the inductive limit of homomorphisms. For more details on the notion of h-inductive sequences and limits, the reader is invited to ([7]).

Parallel to the role of existentially closed structures in the framework of logic with negation, for every h-inductive theory , there exists a class of models of which represent the theory marvellously, and which enjoy the properties desired by every structures; namely, the h-inductive property of the class, the maximality of types (positive formulas satisfied by an element), amalgamation property, and others. These modules are called positively closed.

Definition 1. A model of an h-inductive theory is said to be positively closed (in short, pc) if every homomorphism from to , a model of , is an immersion.
The following lemmas announce the main properties of pc models. They will be used without mention.

Lemma 2 (Lemma 14, see [7]). A model of an h-inductive theory is pc if and only if for every positive formula and , if , then there exists a positive formula such that and .

Lemma 3 (Theorem 2, see [7]). Every model on an h-inductive theory is continued in a pc model of .
For every positive formula , we denote by the set of positive formulas such that

Two h-inductive theories are said to be companion if every model of one of them can be continued into a model of the other or equivalently if the theories have the same pc models.

Every -inductive theory has a maximal companion denoted by called the Kaiser’s hull of . is the set of -inductive sentences satisfied in each pc models of . Likewise, has a minimal companion denoted by , formed by its -universal consequence sentences.

Remark 4. Let and two h-inductive theories. The following propositions are equivalent:(i) and are companion.(ii).(iii).

Definition 5. Let be an h-inductive theory.(i) is said to be model complete if every model of is a pc model of .(ii)We say that has a model companion whenever is a model-complete theory.Let be a model of . We shall use the following notations:(i), the set of positive quantifier-free sentences satisfied by in the language .(ii), the set of atomic and negated atomic sentences satisfied by in the language .(ii)We denote by the -theory .(iv) (resp. ) denote the set of h-inductive (resp. h-universal) -sentences satisfied by .(v) (resp. ) denote the set of h-inductive (resp. h-universal) -sentences satisfied by .(vi) (resp. ) denote the Kaiser’s hull of (resp. of ).(vii)For every subset of , we denote by (resp. ) the set of h-inductive (resp. h-universal) -sentences satisfied by A.

Definition 6. Let and be two -structures and a homomorphism from into . is said to be a strong immersion (in short s-immersion) if is an immersion and is a model of .

Remark 7. Let and two -structures. We have the following properties:(1)If is immersed in , then and .(2) is immersed in if and only if .(3) is strongly immersed in if and only if .(4)If and are two pc models of , then every homomorphism from into is a strong immersion. Indeed, let and be two positive formulas and let the h-inductive sentence . Suppose that and , then there is such that and . Given that is a pc model, there exists such that . Since is immersed in , then there is such that , which implies and , a contradiction.(5)The pc models of the -theory are the pc models of that are the continuation of . Indeed, it is clear that every pc model of in which is continued is a model of and then a pc model of . Conversely, let be a pc model and a pc model of in which is continued by a homomorphism . Then, is a continuation of , so is a model of . Thereby, is an immersion, which implies that is a pc model of .Let and two be -structures and a mapping from into . We will use the following notations:(i): the set homomorphisms from into .(ii): the set embeddings from into .(iii): the set immersions from into .(iv): the set s-immersions from into .

Remark 8. Let and be two -structures and a mapping from into . Consider as a -structure by interpreting the elements of in by . We have the following:(i) if and only if is a model of .(ii) if and only if is a model of .(iii) if and only if is a model of .(iv) if and only if is a model of .

2.2. Positive Amalgamation

To abbreviate the nominations of homomorphism, embedding, immersion, and strong immersion in the definition of the notions of amalgamation, we will use the first letter of each mapping defined above.

Definition 9. Let be a class of -structures and a member of . We say that is -amalgamation basis of , if for every members of , if is continued into by and embedded into by , there exist , , and such that the following diagram commutes:By the same way, we define the notion of -amalgamation property for every .
We say that is an -amalgamation basis of , if is an -amalgamation basis of .
We say that is -symmetric amalgamation basis of whenever is an -amalgamation basis of .
We say that is -asymmetric amalgamation basis of , whenever is an -amalgamation basis of .
The following remark lists some properties of diver forms of amalgamation with the notations and terminology given in the definition above.

Remark 10. (1)Every -structure is an -amalgamation basis in the class of -structures (lemma 4, [6]). Since every strong immersion is an immersion, it follows that every -structure is an -asymmetric amalgamation basis in the class of -structures.(2)Every -structure is an -asymmetric amalgamation basis in the class of -structures (lemma 5, [6]).(3)Every -structure is an -asymmetric amalgamation basis in the class of -structures (lemma 4, [9]).(4)Every -structure is an -asymmetric amalgamation basis in the class of -structures (lemma 8, [7]).(5)Every pc model of an h-inductive theory is an -amalgamation basis in the class of model of .

Lemma 11. Every -structure is -asymmetric amalgamation basis in the class of -structure, where is a homomorphism, an embedding, or an immersion.

Proof. The proof of the lemma directly follows from the Remark 10. More precisely, the cases where is a homomorphism is addressed in bullet 1 of the Remark 10, while the case where is an embedding is covered in bullet 3. The case where is an immersion is addressed in bullet 2.

Lemma 12. A model of is pc if and only if it has the -symmetric amalgamation property in the class of models of .

Proof. Let be an -symmetric amalgamation basis of . Assume that , where and a positive formula. Given that is an -symmetric amalgamation basis, we claim that is inconsistent. Otherwise, we can find two continuations of in which one of them satisfies and the other does not satisfy , which contradicts the assumption that has the -symmetric amalgamation property.

Proposition 13. An h-inductive theory has a model companion if and only if has the - symmetric amalgamation property.

Proof. Suppose that has a model companion, then every model of is a pc model. Since the pc models have the -amalgamation property and the homomorphisms between the pc models are immersion, it follows from the fifth bullet of the Remark 10 that has the -symmetric amalgamation property.
The opposite direction follows easily from Lemma 12.

3. Almost Positively Closed Structures

In this section, we introduce the notions of almost and -almost positively closed models, and we give a syntactic characterization and a characterization via the closed formulas which turns out to be an essential tool in the study of the notion of -almost positively closedness.

Definition 14. Let be an h-inductive theory and a model of . Let be a subset of -quantifier-free positive formulas such that for every , the set is consistent. The model is said to be(i)Almost positively closed (apc in short), if for every model , , and a quantifier-free positive formula, if and , then there is such that .(ii)-almost positively closed (-apc in short), if for every model , , and , if and , then there is such that .(iii)Weakly almost positively closed (wpc in short), if for every pc model , , and a quantifier-free positive formula if and , then there is such that .(iv)-weakly almost positively closed (-wpc in short), if for every pc model , , and , if , then there is such that .

Theorem 15. Let be a model of an h-inductive -theory . Let be a set of -quantifier free positive formula that satisfies the condition of Definition 14. The model is -apc of if and only if for every , there exist and a quantifier-free positive formula such that

Proof. Assume that is an -apc model of and let such that is consistent. Given that is -apc, then is inconsistent. Thus, there are and such thatis inconsistent, which impliesFor the other direction, let be a model of that satisfies the hypothesis of the theorem. Let and , where is a model of and . Given that , then . By the hypothesis of the theorem, we obtain . So, is an -apc model of .

Corollary 16. Let be a model of and a set of quantifier-free positive -formulas. If is immersed in an -apc model of and , then is an -apc model of .

Proof. Let . Given that and is an -apc model of , by Theorem 15, there is such thatOn the other hand, since is immersed in , then there are such that . By Theorem 15, is an -apc of .

Lemma 17. Let be an inductive sequence of models of an h-inductive theory . Suppose that for every , the model is -apc where is a set of quantifier-free positive -formulas such that . Then, the inductive limit of the sequence is -apc of .

Proof. Let be a model of and . Let and such that . Let such that and . Given that where is the canonical homomorphism defined from in , then there is such that . So, is -apc.

Remark 18. Let be an h-inductive -theory and a set of quantifier-free positive -formulas. We have the following properties:(1)If is apc, then is wpc of .(2)Every pc model of is an apc (resp. -apc) model of .(3)The classes of apc and wpc (resp. -apc and -wpc) models of are -inductive.(4)If is an apc model of and a model of , then .(5)Let be two sets of free quantifier positive formulas. If is -apc (resp. -wpc) then is -apc (resp. -wpc).(6)Every apc model of has the property of -asymmetric amalgamation (property 4 of Remark 18, and the property 4 of Remark 10).

Example 1. (1)Let be a functional language. Let be the h-inductive theory. The theory has a model companion axiomatized by . The class of apc model of is elementary and axiomatized by the h-inductive theory.(2)Let and be the functional language and the theory defined in the bullet above. Let the h-inductive theory . The class of apc model of is axiomatized by the h-inductive theory.(3)Let the theory of fields. Since the negation of equality is defined by the positive formula and every homomorphism is an embedding then the classes of apc fields, pc fields, and existentially closed fields are equals.

Definition 19. Let be an h-inductive -theory and a model of .(i)A positive formula is said to be -algebraic if modulo (i.e., has a realisation in some model of ) and there exists a positive formula such that modulo , and We denote by the set of -algebraic quantifier-free positive -formulas.(ii)For every positive formula , we denote by the set of positive formulas such that modulo and satisfy the following property:(iii)A positive formula is said to be -closed if modulo , and for every pc model continuation of , if for some , then .

Remark 20. (1)A quantifier-free positive formula is -algebraic if and only if its algebraic is in the sense of Robinson ([4]).(2)Given that the class of pc models of coincides with the class of pc models of that are continuation of (bullet 5 of Remark 7), then a formula is -closed if and only if it is -closed.(3)Let be a model of . Denote by the set of quantifier-free formulas that are -closed. Then, is -wpc.(4)If every formula in is -closed, by the bullet 2 above, the model is -wpc, and since , is also -wpc.We denote by the set of quantifier-free positive formulas such that .

Lemma 21. Let be an h-amalgamation basis of . If is -wpc (resp. -apc), then every formula in is -closed.

Proof. Let be a -wpc and an h-amalgamation basis of . Assume the existence of a formula such that is not -closed. So, there exist a pc models of and such that . Let , and let be a pc model of and such that . Given that and is an -wpc model of , then there is in such that . Let be a model of that amalgamate commutatively the diagram . Thus, is immersed in and so , which implies , a contradiction.
The proof of the case where is -apc is an immediate.

4. Strong Amalgamation

In this section, we introduce the notions of positive strong amalgamation and h-strong amalgamation. We investigate their properties and interactions with the notion of almost positively closedness.

4.1. Positive Strong Amalgamation

Definition 22. Let be an h-inductive theory. A model of is said to be a positive strong amalgamation basis (in short PSA) (resp. h-strong amalgamation basis (in short h-SA)) of , if for every pc models (resp. models) and of , if is continued into and by two homomorphisms and , respectively, then there exist a model of and , two homomorphisms, such that the following diagram commutes:and satisfies the following property (P):
, if , then there is such that and .

Remark 23. Note that in the definition of PSA basis, we can reformulate the property (P) as follows:
, if , then there exist such that and .
Indeed, let such that and , thenGiven that is an immersion, we have

Example 2. (1)Every h-inductive theory for which the unique pc model is the trivial model has the positive strong amalgamation property. As examples of these theories, we have the theory of groups and the theory of partially ordered sets.(2)Let where and are two unary relation symbols. Let be the h-inductive theory . The trivial structure such that is the unique pc model of ; thus, has the positive strong amalgamation property. However, the structure where has no h-strong amalgamation property. Indeed, let , , and , then the diagram cannot be h-strongly amalgamate.

Lemma 24. Let be three -structures. Let and . Then, there exist a -structure, a homomorphism, and an s-immersion such that the following diagram commutes:and satisfies the following property:
, and if , then there exists such that and .

Proof. The proof consists in the verification that the following set is -consistent.where every elements of is interpreted by the same symbols of constant in and .
Assume that is -inconsistent. Then, there exist , where and such thatGiven that and is immersed in , so there is such that . Consequently, . Thereby, , which is a contradiction.
Let be a model of , then the following digram commutes:where is a homomorphism and a strong immersion. Let and such that , so there is in such that and . By the commutativity of the diagram above, we haveGiven that is an immersion, we obtain

Corollary 25. Every pc model of is a h-strong amalgamation basis of .

Proof. Immediate from Lemma 24.

Proposition 26. Let and be two models of an h-inductive theory . If is immersed in , a h-SA basis of , then is a h-SA basis of .

Proof. Let and be two models of , , and . By applying Lemma 24 to the diagrams and , we get the commutative diagrams (1) and (2), where are homomorphisms and are strong immersions.
Now, given that has the h-strong amalgamation property, we get the commutative diagram (3):where are homomorphisms.
We claim that makes the diagram strongly amalgamate. Indeed, let and such that . By the -strong amalgamation property of the diagram (3), there is such that and . Considering the properties of the diagrams (1) and (2), we get two elements and from such thatGiven that is an immersion, then and . So, is a h-SA basis of .

Lemma 27. An h-amalgamation basis of is a PSA basis if and only if for every pc model of and for every , we have for every .

Proof. Let be a PSA basis of . Suppose that there are and a pc model of such that where . Let and a pc model of such that .
Given that is a PSA basis of , we obtain the following commutative diagram.where and are immersion, and a model of that satisfies the following property:Now, since , then , which implies the existence of an element such that , a contradiction.
For the other direction, let and be two pc models of , , and such that the following diagram is not h-strongly amalgamable.Thus, there exist , where and such thatThen, and .

Theorem 28. Let be an h-amalgamation basis of , then we have the following properties:(1)If is a -wpc model of , then is a PSA basis of .(2)If is a PSA basis of , then is -wpc.

Proof. (1)Let be an h-amalgamation basis and a -wpc model of . Let and be two pc models of , , and . Let a model of such that the following diagram commutes: where and are immersions. We claim that the set is -consistent (note that the element of are interpreted by the same symbols of constants in and ). Assume that the set above is inconsistent. Then there are , and such that is -inconsistent, thereby Now, since , , and is an -wpc model, then there is such that . Thereby, , so , which implies , a contradiction. Thus, is a PSA basis of .(2)Suppose that is PSA of . Since , by Lemma 27, every formula in is -closed, which implies that is a -wpc model of by Remark 20 bullet (4).

Data Availability

No data were used to support the findings of this study.

Disclosure

An earlier version of this article was previously published as preprint on https://arxiv.org/ [10].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The researcher would like to thank the Deanship of Scientific Research, Qassim University, for funding the publication of this project.