Finite Varieties Research Articles

Semilinear De Morgan monoids and epimorphisms

A representation theorem is proved for De Morgan monoids that are (i) semilinear, i.e., subdirect products of totally ordered algebras, and (ii) negatively generated, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a variety—in fact, a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.

Primitive lattice varieties

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition (W), then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with I doubled, then [Formula: see text] is never primitive.

Patterning uncertainty: partial likeness, analogy and likelihood

Abstract This paper examines the role of similarity judgements in Keynes’s theory of analogy as discussed in the Treatise on Probability (Keynes, 1921) and outlines a conceptual framework in which induction by analogy based on the identification of partial resemblances leads to likelihood as the central disposition by which human actors assign a pattern to uncertainty. The structure of the paper is as follows. Section 2 discusses Keynes’ view of analogy as the foundation of inductive inference. Section 3 outlines a theory of similarity based on Keynes’s criteria of finite variety and negative analogy. Section 4 highlights the relationship between a similarity-based theory of induction and Keynes’ ontological view that similarity features derive from a finite number of ‘generator properties’. Section 5 examines the social structuring of similarity features and the active role of agents in constructing the inductive knowledge of the world they inhabit. Section 6 brings the paper to close by calling attention to the relationship between dispositions, circumstantial visualisation of similarity features and problem-solving actions in the world of practice.

Semigroup quasivarieties: Two lattices and a reopened problem

We determine all quasivarieties of aperiodic semigroups that are contained in some residually finite variety. This endeavor was initially motivated by a problem in natural dualities, but our work here also serves as a partial correction to an error found in a result of Sapir from the 1980s.

Max Scheler’s education concept in the light of his philosophical anthropology

In modern pedagogy, the educational ideal of knowledge is gradually being established as a special property of a person included in the existence of the world. The formation of a person comes to the fore here, not just the process of transferring a certain amount of knowledge. One of the founders of this approach is Max Scheler. Materials and methods. The general foundations for the study of his concept of education were the method of philosophical and anthropological reconstruction and the method of meaningful and semantic interpretation, as well as dialectical, hermeneutic, comparative, ascent from the abstract to the concrete, systemic and other methods. The main results of the study. The German thinker developed his theory of education on the basis of philosophical anthropology and the sociology of knowledge, the component parts of which are phenomenology and axiology. The main task of philosophical anthropology is to analyze the essence and essential structure of man, the sources of his being, which combines a powerful but blind energetic-vital “impulse” and a seeing, but weak content-semantic “spirit” axiologically striving towards the Divine. First of all, a person is a loving creature, and then – a knowing and willing. Scheler identified three types of knowledge – practical, positive, aimed at achieving pragmatic goals; essential, educational, aimed at the formation and development of the individual, and metaphysical, salvific, striving to comprehend the higher basis of being, which can be used in the process of education and spiritual development of a person, where the second type of knowledge plays the main role. Education is the process of self-creation of a person through love for other people, God and the whole world, the philosopher emphasizes. The metaphysical prerequisite for the self-formation of a personality is freedom as the spontaneity of its spiritual center. The goal of education is the realization of the ideal of the whole human as a comprehensively and spiritually developed personality, consistently revealing all its essential capabilities. In the absolute sense, the “whole human” is the “guiding idea” of a person who, in infinite form and finite variety, consistently reveals all his essential capabilities. In a relative sense, this ideal is embodied in the idea of a “well-formed human” striving for self-development in specific historical conditions and life circumstances. The comprehensive and holistic worldview of the individual, the philosopher comes to the conclusion, is possible through the acquisition and assimilation of essential, educational and other types of knowledge that determine the structure of ordering and understanding the facts of the possible experience of people.

Leibniz A-algebras

Abstract A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

Semi analytical cum numerical analysis of two-dimensional elasticity problem of plate with eccentric hole in bipolar coordinate system

A semi analytical cum numerical method for analyzing 2D elasticity problem of plate with eccentric hole in bipolar coordinate systems is given. Complete formulation of the plate problem is derived in bipolar coordinate system. A novel numerical integration technique is adopted to obtain the numerical results. Few parameters are identified that adversely affects the stresses and displacements of isotropic and FG plate with eccentric hole. Verification of the mathematical models with numerical results such as finite element models (COMSOL Multiphysics) and wide variety of applicability of the developed models for hypothetical/real field problems is seen from the present work.

Study on new polymer crosslinking agents and their functional hydrogels

Functional hydrogels have been used in various fields, but their development might be limited due to the finite variety of crosslinking agents, complicated functional monomer design and synthesis. To address these issues, a new series of polyamino acid-based multifunctional coupling agents, polyaspartic hydrazide-co-aspartic acid polymers (PHAs), were synthesized through a very simple ring-opening reaction in a short time. The structures of these polymers with gradient content of acylhydrazine and carboxyl groups were characterized through 1H NMR and FT-IR. The molecular weight of PHAs and the content of acylhydrazine and carboxyl groups were tested through GPC and colloidal titration method, respectively. By crosslinking between acylhydrazine groups of PHA crosslinkers and carbonyl of diacetone acrylamide (DAAM), DAPHA hydrogels with different PHA structures and contents were prepared from acrylamide (AAM), DAAM and PHA crosslinkers in a one-step reaction. Furthermore, through coordination between pendant carboxyl of PHAs and metal cations, DAPHA-Fe3+ hydrogels were prepared by soaking DAPHA hydrogels in ferric chloride solution. And the structures and mechanical properties of both DAPHA and DAPHA-Fe3+ hydrogels were investigated. Moreover, taking advantage of the complexation and dissociation of DAPHA hydrogel with Fe3+, their shape memory properties were investigated. Finally, magnetic DAPHA-Fe3O4 hydrogels were prepared through in-situ precipitation method, and their magnetism has been studied.

Approaching Other Animals with Caution: Exploring Insights from Aquinas's Psychology

AbstractIn this essay I explore the resources Thomas Aquinas provides for enquiries concerning the psychological abilities of nonhuman animals. I first look to Aquinas's account of divine, angelic, human, and nonhuman animal naming, to help us articulate the contours of a ‘critical anthropocentrism’ that aims to steer clear of the mistakes of a naïve anthropocentrism and misconceived avowals to entirely eschew anthropocentrism. I then address the need for our critical anthropocentrism both to reject the mental-physical dichotomy endorsed by ‘folk psychology’ and to articulate a more adequate ‘commonsense psychology’ that acknowledges most embodied animal behavior is observable psychological behavior. Next, I argue that we can develop Aquinas's doctrine of estimation and conation to formulate an account of nonhuman animal action that more adequately characterizes the purposeful behaviors of nonhuman animals. To do so, we first need to recognize a wider range of nonhuman animal behaviors that are captured by Aquinas's ‘estimative sense’, and that all of these behaviors are specified by a finite variety of particular goods confined to the animals’ environmental niches. But we also need to supplement Aquinas's account of human and nonhuman animal agency by exploring the ontogeny and ecology of how humans and other animals become attuned to affordances within these different environmental niches. I argue that we should look to Aquinas's account of nonhuman animal capacities in ST I-II 6-17 for subtle insights that can expand our understanding of how nonhuman animals engage in purposeful behavior by exercising analogous nonrational and imperfectly voluntary forms of intention, deliberation, choice, execution, and enjoyment. I conclude with an outline for how future enquiry can seek to explain the nonrational purposeful problem-solving competencies of chimps, canines, corvids, cetaceans, cephalopods, and other nonhuman animal species.

A novel inhomogeneous deformation field model for cylindrical specimens for unconfined compression testing

Abstract Compression methods when testing polymer materials from quasi-static to dynamic loading rates require the use of cylindrical specimens due symmetry. For soft materials, which have more complex non-linear response such as hyperelastic or viscoelastic compared to traditional engineering materials, additional challenges arise when obtaining their material parameters. Previous studies have been found scarce in investigating and developing mathematical models to predict this complex deformation field whenever friction is considered during unconfined compression testing. We present an analytical approach to model this “barreling effect” that is derived from inhomogeneous deformation field that accumulates mass in the center region of the specimen. This reduces the contact surface area that directly affects the computation of material constants e.g. stress. Our novel method successfully describes this effect quantitatively, providing an analytical solution that models the deformation field for a wide range of finite strains and variety materials. This approach has been experimentally proved to dramatically increase the accuracy by up to 17% compared to existing methodologies that are based only on the homogeneous deformation approach, which to date, is the most common model to characterize the general material's mechanical response.

Generalized Eilenberg Theorem

For finite automata as coalgebras in a category C , we study languages they accept and varieties of such languages. This generalizes Eilenberg’s concept of a variety of languages, which corresponds to choosing as C the category of Boolean algebras. Eilenberg established a bijective correspondence between pseudovarieties of monoids and varieties of regular languages. In our generalization, we work with a pair C / D of locally finite varieties of algebras that are predual, i.e., dualize on the level of finite algebras, and we prove that pseudovarieties of D -monoids bijectively correspond to varieties of regular languages in C . As one instance, Eilenberg’s result is recovered by choosing D = sets and C = Boolean algebras. Another instance, Pin’s result on pseudovarieties of ordered monoids, is covered by taking D = posets and C = distributive lattices. By choosing as C amp;equals; D the self-predual category of join-semilattices, we obtain Polák’s result on pseudovarieties of idempotent semirings. Similarly, using the self-preduality of vector spaces over a finite field K , our result covers that of Reutenauer on pseudovarieties of K -algebras. Several new variants of Eilenberg’s theorem arise by taking other predualities, e.g., between the categories of non-unital Boolean rings and of pointed sets. In each of these cases, we also prove a local variant of the bijection, where a fixed alphabet is assumed and one considers local varieties of regular languages over that alphabet in the category C .

Towards a pseudoequational proof theory

A new scheme for proving pseudoidentities from a given set \Sigma of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when \Sigma defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples for which the scheme is complete are given when \Sigma defines a pseudovariety V which is \sigma -reducible for the equation x = y , provided \Sigma is enough to prove a basis of identities for the variety of \sigma -algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.

The Ramsey and the Ordering Property for Classes of Lattices and Semilattices

The class of finite distributive lattices, as many other classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokic have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the variety of distributive lattices is not an exception, but an instance of a more general phenomenon. We show that for almost all nontrivial locally finite varieties of lattices no “reasonable” expansion of the finite members of the variety by linear orders gives rise to a Ramsey class. The responsibility for this lies not with the lattices as structures, but with the lack of algebraic morphisms: if we consider lattices as partially ordered sets (and thus switch from algebraic embeddings to embeddings of relational structures) we show that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property. It now comes as no surprise that the same is true for varieties of semilattices.

A characterization of idempotent strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties

We prove a characterization of all idempotent, linear, strong Mal’cev conditions in two variables which hold in all locally finite congruence meet-semidistributive varieties. This is an alternative proof to the one previously given by Z. Brady, and has some advantages, some disadvantages, to his approach. Along the way we prove that such a strong Mal’cev condition holds in all locally finite congruence meet-semidistributive varieties iff it is realized in a certain four-element algebra.

Congruence meet-semidistributive locally finite varieties and a finite basis theorem

We provide several conditions that, among locally finite varieties, characterize congruence meet-semidistributivity and we use these conditions to give a new proof of a finite basis theorem published by Baker, McNulty, and Wang in 2004. This finite basis theorem extends Willard’s Finite Basis Theorem.

Quasiorder lattices of varieties

The set $${{\mathrm{Quo}}}(\mathbf {A})$$ of compatible quasiorders (reflexive and transitive relations) of an algebra $$\mathbf {A}$$ forms a lattice under inclusion, and the lattice $${{\mathrm{Con}}}(\mathbf {A})$$ of congruences of $$\mathbf {A}$$ is a sublattice of $${{\mathrm{Quo}}}(\mathbf {A})$$ . We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice $$\mathbf {M}_3$$ . We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.

Simpler Maltsev conditions for (weak) difference terms in locally finite varieties

This paper is motivated by a practical question: given a finite algebra A in a finite language, how can we best program a computer to decide whether the variety generated by A has a difference term, and how hard is it to find the difference term? To help address this question, we produce a simple Maltsev condition which characterizes difference terms in the class of locally finite varieties. We do the same for weak difference terms.

Epimorphisms in varieties of residuated structures

It is proved that epimorphisms are surjective in a range of varieties of residuated structures, including all varieties of Heyting or Brouwerian algebras of finite depth, and all varieties consisting of Gödel algebras, relative Stone algebras, Sugihara monoids or positive Sugihara monoids. This establishes the infinite deductive Beth definability property for a corresponding range of substructural logics. On the other hand, it is shown that epimorphisms need not be surjective in a locally finite variety of Heyting or Brouwerian algebras of width 2. It follows that the infinite Beth property is strictly stronger than the so-called finite Beth property, confirming a conjecture of Blok and Hoogland.

Symmetric operators on modal pseudocomplemented De Morgan algebras

Modal pseudocomplemented De Morgan algebras are pseudocomplemented De Morgan algebras which satisfy the additional equation |$\mbox{(tm) }\, x\vee \sim x \leq x\vee x^\ast.$| There is a connection between these algebras and A. Monteiro’s tetravalent modal algebras. It is worth mentioning that these algebras have been studied previously using topological tools. In the present work, we enrich the language of modal pseudocomplemented De Morgan algebras by adding an automorphism of period |$2$|⁠, giving rise to a new algebraic structure which we call |$\cal S$|-algebras. From the logical point of view, we are enriching the original language with a new involutive unary connective. Among other things, it is proved that |$\cal S$| is a semisimple, locally finite variety. Besides, the generating algebras of this variety are determined. The structure of the free |$\cal S$|-algebra with a finite number of generators is studied, and a formula for computing its size is given. Finally, the lattice |$\Lambda({\cal S})$| is exhibited showing that the varieties of Boolean algebras, involutive Boolean algebras and three-valued Łukasiewicz algebras are proper subvarieties of |${\cal S}$|⁠.

Optimal strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties

Locally finite, congruence meet-semidistributive varieties have been characterized by numerous Mal’cev conditions and, recently, by two strong Mal’cev conditions. We provide three new strong Mal’cev characterizations and a new Mal’cev characterization each of which improves the known ones in some way.