Amalgamation Property Research Articles

Limit models in strictly stable abstract elementary classes

AbstractIn this paper, we examine the locality condition for non‐splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that is an abstract elementary class satisfying the joint embedding and amalgamation properties with no maximal model of cardinality , stability in , , continuity for (i.e., if and is a limit model witnessed by for some limit ordinal and there exists so that does not ‐split over for all , then does not ‐split over ). Then for and limit ordinals both with cofinality , if satisfies symmetry for (or just ‐symmetry), then, for any and that are and ‐limit models over , respectively, we have that and are isomorphic over . Note that no tameness is assumed.

Root-filling materials for endodontic surgery: biological and clinical aspects

The placement of root filling materials aims to prevent the occurrence of post-treatment apical periodontitis following completion of endodontic treatment. Materials should possess properties that will not permit bacterial invasion and infection, namely excellent sealing ability and/or antibacterial properties. In root-end filling procedures or repair of root perforations, the root filling materials are placed in a particularly challenging clinical environment, as they interface with a relatively large area with the periradicular tissues. The biological properties of these materials are therefore of significant importance. The current review discusses the most widely used materials for endodontic surgery (i.e., root-end filling and perforation repair), with particular focus on their biological characteristics, namely antibacterial properties and interactions with host tissue cells, together with clinical studies. Properties of amalgam, glass ionomer cements (GICs), resin systems, zinc oxide eugenol-based cements and hydraulic calcium silicate cements (HCSCs), together with representative and well-researched commercial materials in the context of their use in endodontic surgery are presented. While the use of HCSCs seems to offer several biological advantages, together with addressing issues with the initial formulation in the most recent versions, materials with different chemical compositions, such as zinc oxide eugenol-based cements, are still in use and appear to provide similar clinical success rates to HCSCs. Thus, the significance of the currently available materials on clinical outcomes remains unclear.

Random structures and automorphisms with a single orbit

AbstractWe investigate the class of m-hypergraphs whose substructures with l elements have more than sm-element subsets that do not form a hyperedge. The class will have the free amalgamation property if s is small, but it does not if s is large. We find the boundary of s. Suppose the class has the free amalgamation property. In the case $$m \ge 3$$ m ≥ 3 , we demonstrate that the random structure for the class has continuum-many automorphisms with a single orbit. The situation differs from the case of Henson graphs. In the case of generic hypergraphs constructed by Hrushovski’s method using a predimension function, we also demonstrate that they have no automorphisms with a single orbit.

Adding an implication to logics of perfect paradefinite algebras

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion () order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Effect of stirring time on viscoelastic properties of liquid gallium-oxide amalgams

Effect of stirring time on viscoelastic properties of liquid gallium-oxide amalgams

Positive Complete Theories and Positive Strong Amalgamation Property

We introduce the notion of positive strong amalgamation property and we investigate some universal forms and properties of this notion. Considering the close relationship between the amalgamation property and the notion of complete theories, we explore the fundamental properties of positively complete theories, and we illustrate the behaviour of this notion by bringing changes to the language of the theory through the groups theory.

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.

ON THE STRUCTURE OF BOCHVAR ALGEBRAS

Abstract Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$ . Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP).

Almost Existentially Closed Models in Positive Logic

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 Δ-almost positively closed and Δ-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 ET and AlgT and the properties of positive strong amalgamation. We then show that if a model A of a theory T is a ET+A-weekly almost positively closed, then A is a positive strong amalgamation basis of T, and if A is a positive strong amalgamation basis of T, then A is AlT+A-weekly almost positively closed.

The Structure of Semiconic Idempotent Commutative Residuated Lattices

In this paper, we study semiconic idempotent commutative residuated lattices. After giving some properties of such residuated lattices, we obtain a structure theorem for semiconic idempotent commutative residuated lattices. As an application, we make use of the structure theorem to prove that the variety of strongly semiconic idempotent commutative residuated lattices has the amalgamation property.

Preservation of NATP

We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of [Formula: see text]-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved. Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP1 and NTP2 is considered. Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.

Transfer theorems for finitely subdirectly irreducible algebras

We show that under certain conditions, well-studied algebraic properties transfer from the class QRFSI of the relatively finitely subdirectly irreducible members of a quasivariety Q to the whole quasivariety, and, in certain cases, back again. First, we prove that if Q is relatively congruence-distributive, then it has the Q-congruence extension property (Q-CEP) if and only if QRFSI has this property. We then prove that if Q has the Q-CEP and QRFSI is closed under subalgebras, then Q has a one-sided amalgamation property (for quasivarieties, equivalent to the amalgamation property) if and only if QRFSI has this property. We also establish similar results for the transferable injections property and strong amalgamation property. For each property considered, we specialize our results to the case where Q is a variety — so that QRFSI is the class of finitely subdirectly irreducible members of Q and the Q-CEP is the usual congruence extension property — and prove that when Q is finitely generated and congruence-distributive, and QRFSI is closed under subalgebras, possession of the property is decidable. Finally, as a case study, we provide a complete description of the subvarieties of a notable variety of BL-algebras that have the amalgamation property.

Higher amalgamation properties in measured structures

Higher amalgamation properties in measured structures

Predesigned covalent organic framework with sulfur coordination: Anchoring Au nanoparticles for sensitive colorimetric detection of Hg(II)

Targeted construction of new covalent organic frameworks (COFs) with specific purposes and rationalities to build colorimetric assay platform for environmental pollutant monitoring have attracted increasing interest. However, it is still challenging due to lack of available coordination sites inside COFs pores and only a slight bonding ability for anchoring metal. In this work, a two-dimensional (2D) COFs (termed as Tz-COF) with high crystallinity, excellent chemical stability, and abundant sulfur coordination in its skeletons was synthesized and used for the confined growth of Au NPs. It was found that the Au NPs showed significant dispersibility for the support of Tz-COF. The proposed Tz-COF@Au NPs possessed outstanding Hg2+-activated peroxidase-like activity benefited from physicochemical properties of gold amalgam and synergistic effect between COFs and Au NPs to oxidize chromogenic substrate. Based on highly efficient activity and distinctive color evolution, the strategy for detecting Hg2+ was developed and successfully applied to determine the content of Hg2+ in real environmental samples. This work manifests that a potential strategy to establish a colorimetric assay platform for environmental pollutant monitoring based on the targeted manufacturing of novel COFs with specific functions.

Interpolation Results for Arrays with Length and MaxDiff

In this paper, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. As is well-known, some diff operation (i.e., some kind of difference function showing where two unequal array differ) is needed to keep interpolants quantifier-free in array theories; our maxdiff operation returns the max index where two arrays differ and so it has a univocally determined semantics. The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays. Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory and we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols. In this paper, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. It is known from the literature that a diff operation is required in order for the theory of arrays to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom): instead, our maxdiff operation returns the max index where two arrays differ and so it is univocally determined at the semantic level. The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays (which are defined in their specified allocation memory). Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory and we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols. The array theory in the paper has been modified so as to model real arrays used in common programming languages (we now require them to be ‘contiguous’, i.e., not undefined in any of their allocation entries); moreover strong amalgamation and interpolation with free function symbols are proved. The interpolation algorithm avoids full instantiation routines and unbounded loops, thus achieving the above mentioned polynomial reduction complexity. The present paper is a substantially revised version of a previous conference paper presented at FoSSaCS 2021.

Some definable types that cannot be amalgamated

AbstractWe exhibit a theory where definable types lack the amalgamation property.

NSOP-LIKE INDEPENDENCE IN AECATS

AbstractThe classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing.

Amalgamation and Robinson property in universal algebraic logic

Abstract There is a well-established correspondence between interpolation and amalgamation for algebraizable logics that satisfy certain additional assumptions. In this paper, we introduce the Robinson property of a logic and show that a conditionally algebraizable logic without any additional assumptions has the Robinson property if and only if the corresponding class of Lindenbaum–Tarski algebras has the amalgamation property. Moreover, we give the logical characterization of the strong amalgamation property, solving an open problem of Andréka–Németi–Sain. It is also shown that given the mentioned extra assumptions the Robinson property implies the interpolation property. As conditionally algebraizable logics cover algebraizable logics as well as various quantifier logics such as classical first order logic, our results yield a generalization of some of the results concerning interpolation and amalgamation.

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.

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.