Universal Algebra Research Articles

New Fundamental Relation on Fuzzy Hypersemigroups

The important implements in the fuzzy hyperstructure theory are fundamental relations to acquire universal algebras. The fundamental relation on a fuzzy hypersemigroup(hypergroup) is introduced as the smallest equivalence relation such that the factor would be a semigroup (group). In this study, a novel fuzzy strongly regular relation on a fuzzy hypersemigroup (hypergroup) is characterized so that the quotient is a commutative semigroup (group). The necessary and sufficient circumstances are expressed in which the given relation is transitive.

On ultrafilter extensions of first-order models and ultrafilter interpretations

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jonsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Muller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Cech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Muller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456 ). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017).

Alpha Ideals and the Space of Prime Alpha Ideals in Universal Algebras

The purpose of this paper is to study α -ideals in a more general context, in universal algebras having a constant 0 . Several characterizations are obtained for an ideal I of an algebra A to be an α -ideal. It is shown that the class of all α -ideals of an algebra A forms an algebraic lattice. Prime α -ideals and several related properties are investigated. Some properties of the spectral space of prime α -ideals equipped with the hull-kernel topology are derived.

L-fuzzy cosets in universal algebras

AbstractIn this paper, we introduce the notion of fuzzy costs in a more general context, in universal algebra by the use of coset terms. We study the structure of fuzzy cosets by applying the general theory of algebraic fuzzy systems. Fuzzy cosets generated by a fuzzy set are characterized in different ways. It is also proved that the class of fuzzy cosets determined by an element forms an algebraic closure fuzzy set system. Finally, we give a set of necessary and sufficient conditions for a given variety of algebras to be congruence permutable by applying the theory of fuzzy cosets.

Free objects and Gröbner-Shirshov bases in operated contexts

This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set Φ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free Φ-algebras, via universal algebra. Free Φ-algebras over algebras are studied in details. A mild sufficient condition is found such that Φ together with a Gröbner-Shirshov basis of an algebra A form a Gröbner-Shirshov basis of the free Φ-algebra over algebra A in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI.

Metric monads

Abstract We develop universal algebra over an enriched category and relate it to finitary enriched monads over . Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.

Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

Constructing Higher Inductive Types as Groupoid Quotients

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. Then we construct a biinitial object in the bicategory of algebras in groupoids, which gives the desired algebra. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. We present several examples of HITs which are definable using our notion of signature. In particular, we show that each signature gives rise to a HIT corresponding to the freely generated algebraic structure over it. We also start the development of universal algebra in 1-types. We show that the bicategory of algebras has PIE limits, i.e. products, inserters and equifiers, and we prove a version of the first isomorphism theorem for 1-types. Finally, we give an alternative characterization of the foundamental groups of some HITs, exploiting our construction of HITs via the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

A conjecture concerning the q-Onsager algebra

The q-Onsager algebra Oq is defined by two generators W0,W1 and two relations called the q-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for Oq with elements denoted{Bnδ+α0}n=0∞,{Bnδ+α1}n=0∞,{Bnδ}n=1∞. In their recent study of a current algebra Aq, Baseilhac and Belliard conjecture that there exist elements{W−k}k=0∞,{Wk+1}k=0∞,{Gk+1}k=0∞,{G˜k+1}k=0∞ in Oq that satisfy the defining relations for Aq. In order to establish this conjecture, it is desirable to know how the elements on the second displayed line above are related to the elements on the first displayed line above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, a proof of the analog conjecture for the Onsager algebra O, and a proof of our conjecture for a homomorphic image of Oq called the universal Askey-Wilson algebra.

Derived gluing construction of chiral algebras

We discuss the gluing construction of class $\mathcal{S}$ chiral algebras in derived setting. The gluing construction in non-derived setting was introduced by Arakawa to construct a family of vertex algebras of which the associated varieties give genus zero Moore-Tachikawa symplectic varieties. Motivated by the higher genus case, we introduce a dg vertex algebra version $\mathsf{MT}_{\mathrm{ch}}$ of the category of Moore-Tachikawa symplectic varieties, where a morphism is given by a dg vertex algebra equipped with action of the universal affine vertex algebra, and composition of morphisms is given by the BRST reduction. We also show that the procedure taking the associated scheme of gives a functor from $\mathsf{MT}_{\mathrm{ch}}$ to the category $\mathsf{MT}$ of derived Moore-Tachikawa varieties, which would imply compatibility of gluing constructions in both categories.

Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems

The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups.

Γ∗-Relation on Fuzzy Hyperrings and Fundamental Ring

Fundamental relation performs an important role on fuzzy algebraic hyperstructure and is considered as the smallest equivalence relation such that the quotient is a universal algebra. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings such that the set of the quotient is a ring that is non-commutative. Also, we introduce the concept of a complete part of a fuzzy hyperring and study its principal traits. At last, we convey the relevance between the fundamental relation and complete parts of a fuzzy hyperring.

Categorification of algebraic quantum field theories

This paper develops a concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are constructed by a local gauging construction for finite groups, which admits a physical interpretation in terms of orbifold theories. A categorification of Fredenhagen’s universal algebra is developed and also computed for simple examples of 2AQFTs.

The universal associative enveloping algebra of a Lie–Jordan algebra with a unit

We describe the universal associative enveloping algebra of a Lie–Jordan algebra with a unit over a field of characteristic 0. We show that if L is a Lie–Jordan algebra with a unit e over a field of characteristic 0, then the (unital) universal associative envelope of L is isomorphic to where The center of the universal associative envelope is also described. We conclude the paper with some open problems for further research.

The Relation “Commutator Equals Function” in Banach Algebras

The relation $xy-yx=h(y)$, where $h$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of this equality, we assume that $x$ and $y$ are elements of a Banach algebra (or of an Arens--Michael algebra). We prove that the universal algebra generated by a commutation relation of this kind can be represented explicitly as an analytic Ore extension. An analysis of the structure of the algebra shows that the set of holomorphic functions of $y$ degenerates, but at each zero of $h$, some local algebra of power series remains. Moreover, this local algebra depends only on the order of the zero. As an application, we prove a result about closed subalgebras of holomorphically finitely generated algebras.

Hereditary Torsion Theories for Graphs

Using congruences, a Hoehnke radical can be defined for graphs in the same way as for universal algebras. This leads in a natural way to the connectednesses and disconnectednesses (= radical and semisimple classes) of graphs. It thus makes sense to talk about ideal-hereditary Hoehnke radicals for graphs (= hereditary torsion theories). All such radicals for the category of undirected graphs which allow loops are explicitly determined. Moreover, in contrast to what is the case for the well-known algebraic categories, it is shown here that such radicals for graphs need not be Kurosh–Amitsur radicals.

Anticommutativity and the triangular lemma

For a variety $${\mathcal {V}}$$ , it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points $$\pi : \mathrm {Pt}({\mathbb {C}}) \rightarrow {\mathbb {C}}$$ , if and only if Gumm’s shifting lemma holds on pullbacks in $${\mathcal {V}}$$ . In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain categorical anticommutativity condition. In particular, we show that this anticommutativity and its local version are Mal’tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corollary, every locally anticommutative variety $${\mathcal {V}}$$ has directly decomposable congruence classes in the sense of Duda, and the converse holds if $${\mathcal {V}}$$ is idempotent.

MALTSEV CONDITIONS FOR GENERAL CONGRUENCE MEET-SEMIDISTRIBUTIVE ALGEBRAS

AbstractMeet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev condition characterizing the meet-semidistributive varieties, and provide a candidate for a strong Maltsev condition.

Weakly Jauch–Piron States in Effect Algebras

We study various types of weakly Jauch–Piron states and various types of state-space properties in effect algebras. We generalize several results of Tkadlec (Tatra Mt. Math. Publ. 10, 55–62 1997; Algebra Universalis 61, 187–194 2009) and of Matousek and Ptak (Internat. J. Theoret. Phys. 54, 4476–4481 2015). In particular, we show when an effect algebra is an orthomodular poset, when a unital set of states is strongly order determining, and we present some state space characterizations of Boolean algebras.

Simplicity of vacuum modules and associated varieties

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.