Joint Embedding Property Research Articles

Varieties defined by basic equations have the amalgamation property

A variety is a class of algebraic structures axiomatized by a set of equations. An equation is basic if there is at most one occurrence of a nonnullary operation symbol on each side. We show that a variety axiomatized by basic equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds, hence if the language is finite we get the existence of Fraïssé limits for finite algebras. Examples include virtually all the varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operation symbols appear in the equations.

Hanf number of the first stability cardinal in AECs

We show that ℶ(2LS(K))+ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, and (<ℵ0)-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties insofar as it allows the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any μ<ℶ(2LS(K))+. Boney conjectured that the joint embedding property is needed for two type-counting lemmas. We solved the conjecture by showing it is independent of ZFC.

Structural Completeness in Many-Valued Logics with Rational Constants

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.

Existentially positive Mustafin theories of S-acts over a group

The paper is connected with the study of Jonsson spectrum notion of the fixed class of models of Sacts signature, assuming a group as a monoid of S-acts. The Jonsson spectrum notion is effective when describing theoretical-model properties of algebras classes whose theories admit joint embedding and amalgam properties. It is usually sufficient to consider universal-existential sentences true on models of this class. Up to the present paper, the Jonsson spectrum has tended to deal only with Jonsson theories. The authors of this study define the positive Jonsson spectrum notion, the elements of which can be, non-Jonsson theories. This happens because in the definition of the existentially positive Mustafin theories considered in a given paper involve not only isomorphic embeddings, but also immersions. In this connection, immersions are considered in the definition of amalgam and joint embedding properties. The resulting theories do not necessarily have to be Jonsson. We can observe that the above approach to the Jonsson spectrum study proves to be justified because even in the case of a non-Jonsson theory there exists regular method for finding such Jonsson theory that satisfies previously known notions and results, but that will also be directly related to the existentially positive Mustafin theory in question.

Universal Horn Sentences and the Joint Embedding Property

The finite models of a universal sentence $\Phi$ in a finite relational signature are the age of a structure if and only if $\Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $\Phi$ has the joint embedding property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most two.

Connection between the amalgam and joint embedding properties

The paper aims to study the model-theoretic properties of differentially closed fields of zero and positive characteristics in framework of study of Jonsson theories. The main attention is paid to the amalgam and joint embedding properties of DCF theory as specific features of Jonsson theories, namely, the implication of JEP from AP. The necessity is identified and justified by importance of information about the mentioned properties for certain theories to obtain their detailed model-theoretic description. At the same time, the current apparatus for studying incomplete theories (Jonsson theories are generally incomplete) is not sufficiently developed. The following results have been obtained: The subclasses of Jonsson theories are determined from the point of view of joint embedding and amalgam properties. Within the exploration of one of these classes, namely the AP-theories, that the theories of differential and differentially closed fields of characteristic 0, differentially perfect and differentially closed fields of fixed positive characteristic are shown to be Jonsson and perfect. Along with this, the theory of differential fields of positive characteristic is considered as an example of an AP-theory that is not Jonsson, but has the model companion, which is perfect Jonsson theory, and the sufficient condition for the theory of differential fields is formulated in the context of being Jonsson.

Examples of weak amalgamation classes

AbstractWe present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum‐sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure M, Polish group G of permutations of M, and n ge 1, G has a comeager n-diagonal conjugacy class iff the family of all n-tuples of G-extendable bijections between finitely generated substructures of M, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate 1- and 2-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.

Atomicity and Well Quasi-Order for Consecutive Orderings on Words and Permutations

Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are (a) being atomic, i.e., not being decomposable as a union of two downward closed proper subsets or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are (1) words over a finite alphabet under the consecutive subword ordering, and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are characterizations of atomicity and well quasi-order for the subpath ordering on paths of a finite directed graph.

On universal modules with pure embeddings

AbstractWe show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first‐order theory with an infinite model extending the theory of R‐modules and (where ⩽pp stands for “pure submodule”). Assume has the joint embedding and amalgamation properties. If or , then has a universal model of cardinality λ. As a special case, we get a recent result of Shelah [28, 1.2] concerning the existence of universal reduced torsion‐free abelian groups with respect to pure embeddings.We begin the study of limit models for classes of R‐modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure‐injective and we characterize limit models with chains of countable cofinality. This can be used to answer [18, Question 4.25].As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

Singly generated quasivarieties and residuated structures

AbstractA quasivariety of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if is relatively semisimple then it is generated by one ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics.

AMALGAMABLE DIAGRAM SHAPES

AbstractA category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category I, the following are equivalent: (i) every I-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every I-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category ${\cal L}\left( {\bf{I}} \right)$ of “paths” in I has only idempotent endomorphisms. When I is a finite poset, these are further equivalent to: (iv) every upward-closed subset of I is simply-connected; (v) I can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite I.

Countable infinite existentially closed models of universally axiomatizable theories

In the present article, we obtain a new criterion for amodel of a universally axiomatizable theory to be existentially closed. The notion of a maximal existential type is used in the proof and for investigating properties of countable infinite existentially closed structures. The notions of a prime and a homogeneous model, which are classical for the general model theory, are introduced for such structures. We study universal theories with the joint embedding property admitting a single countable infinite existentially closed model. We also construct, for every natural n, an example of a complete inductive theory with a countable infinite family of countable infinite models such that n of them are existentially closed and exactly two are homogeneous.

The joint embedding property and maximal models

We introduce the notion of a `pure' Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.34 and Corollary 3.38): If $$\langle \lambda _i: i\le \alpha <\aleph _1\rangle $$??i:i≤?<?1? is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$$(<\lambda _0)$$(<?0), there is an $$L_{\omega _1,\omega }$$L?1,?-sentence $$\psi $$? whose models form a pure AEC and(1)The models of $$\psi $$? satisfy JEP$$(<\lambda _0)$$(<?0), while JEP fails for all larger cardinals and AP fails in all infinite cardinals.(2)There exist $$2^{\lambda _i^+}$$2?i+ non-isomorphic maximal models of $$\psi $$? in $$\lambda _i^+$$?i+, for all $$i\le \alpha $$i≤?, but no maximal models in any other cardinality; and(3)$$\psi $$? has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number $$\aleph _0$$?0 are at least $$\beth _{\omega _1}$$??1. We show that although $$AP(\kappa )$$AP(?) for each $$\kappa $$? implies the full amalgamation property, $$JEP(\kappa )$$JEP(?) for each $$\kappa $$? does not imply the full joint embedding property. We prove the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.

Symmetry and the union of saturated models in superstable abstract elementary classes

Our main result (Theorem 1) suggests a possible dividing line (μ-superstable + μ-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. Theorem 1LetKbe an abstract elementary class with no maximal models of cardinalityμ+which satisfies the joint embedding and amalgamation properties. Supposeμ≥LS(K). IfKis μ- andμ+-superstable and satisfiesμ+-symmetry, then for any increasing sequence〈Mi∈K≥μ+|i<θ<(sup⁡‖Mi‖)+〉ofμ+-saturated models,⋃i<θMiisμ+-saturated. We also apply results of [18] and use towers to transfer symmetry from μ+ down to μ in abstract elementary classes which are both μ- and μ+-superstable: Theorem 2SupposeKis an abstract elementary class satisfying the amalgamation and joint embedding properties and thatKis both μ- andμ+-superstable. IfKhas symmetry for non-μ+-splitting, thenKhas symmetry for non-μ-splitting.

A Note on Saturated Models for Many-Valued Logics

Abstract In this short paper we will discuss on saturated and κ-saturated models of many-valued (t-norm based fuzzy) logics. Using these peculiar structures we show a representation theorem à la Di Nola for several classes of algebras including MV, Gödel, product, BL, NM and WNM-algebras. Then, still using (κ)-saturated algebras, we finally show that some relevant subclasses of algebras related to many-valued logics also enjoy the joint embedding property and the amalgamation property.

ALMOST GALOIS ω-STABLE CLASSES

AbstractTheorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.

Homomorphic Image Orders on Combinatorial Structures

Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial structures. We focus on three aspects: antichains and partial well-order, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the well-known joint embedding property and amalgamation property, and are investigated here for the first time.

On the homogeneous countable Boolean contact algebra

In a recent paper, we have shown that the class of Boolean contact algebras (BCAs) has the hereditary property, the joint embedding property and the amalgamation property. By Fraisse’s theorem, this shows that there is a unique countable homogeneous BCA. This paper investigates this algebra and the relation algebra generated by its contact relation. We first show that the algebra can be partitioned into four sets {0}, {1}, K, and L, which are the only orbits of the group of base automorphisms of the algebra, and then show that the contact relation algebra of this algebra is finite, which is the first non-trivial extensional BCA we know which has this property.

Small permutation classes

We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley–Wilf limit) less than κ but uncountably many permutation classes of growth rate κ, answering a question of Klazar. We go on to completely characterize the possible sub-κ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, the substitution decomposition, and atomicity (a.k.a. the joint embedding property).