Universal Algebra Research Articles

On Algebraic and More General Categories Whose Split Epimorphisms Have Underlying Product Projections

We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of semi-abelian, protomodular, unital and subtractive varieties as well as varieties of right Ω-loops and biternary systems.

Isomorphism is equality

The setting of this work is dependent type theory extended with the univalence axiom. We prove that, for a large class of algebraic structures, isomorphic instances of a structure are equal—in fact, isomorphism is in bijective correspondence with equality. The class of structures includes monoids whose underlying types are “sets”, and also posets where the underlying types are sets and the ordering relations are pointwise “propositional”. For monoids on sets equality coincides with the usual notion of isomorphism from universal algebra, and for posets of the kind mentioned above equality coincides with order isomorphism.

Q-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

We study a relationship between Q-polynomial distance-regular graphs and the double affine Hecke algebra of type (C1∨,C1). Let Γ denote a Q-polynomial distance-regular graph with vertex set X. We assume that Γ has q-Racah type and contains a Delsarte clique C. Fix a vertex x∈C. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a C-vector space W.The universal double affine Hecke algebra of type (C1∨,C1) is the C-algebra Hˆq defined by generators {tn±1}n=03 and relations (i) tntn−1=tn−1tn=1; (ii) tn+tn−1 is central; (iii) t0t1t2t3=q−1/2. In this paper, we display an Hˆq-module structure for W. For this module and up to affine transformation,•t0t1+(t0t1)−1 acts as the adjacency matrix of Γ;•t3t0+(t3t0)−1 acts as the dual adjacency matrix of Γ with respect to C;•t1t2+(t1t2)−1 acts as the dual adjacency matrix of Γ with respect to x. To obtain our results we use the theory of Leonard systems.

Lie algebras in symmetric monoidal categories

We study the algebras that are defined by identities in the symmetric monoidal categories; in particular, the Lie algebras. Some examples of these algebras appear in studying the knot invariants and the Rozansky-Witten invariants. The main result is the proof of the Westbury conjecture for a K3-surface: there exists a homomorphism from a universal simple Vogel algebra into a Lie algebra that describes the Rozansky-Witten invariants of a K3-surface. We construct a language that is necessary for discussing and solving this problem, and we formulate nine new problems.

FUZZY EQUATIONAL CLASSES ARE FUZZY VARIETIES

In the framework of fuzzy algebras with fuzzy equalities and a complete lattice as a structure of membership values, we investigate fuzzy equational classes. They consist of special fuzzy algebras fullling the same fuzzy identities, dened with respect to fuzzy equalities. We introduce basic notions and the corresponding operators of universal algebra: construction of fuzzy subalgebras, homomorphisms and direct products. We prove that every fuzzy equational class is closed under these three operators, which means that such a class is a fuzzy variety.

Hagemann’s theorem for regular categories

In this paper we extend the characterisation of $$n$$ -permutable varieties of universal algebras due to J. Hagemann to regular categories. In particular, we show that a regular category has $$n$$ -permutable congruences if and only if every internal reflexive relation $$R$$ in it satisfies $$R^\circ \leqslant R^{n-1}$$ , and if and only if every internal reflexive relation $$R$$ in it satisfies $$R^n\leqslant R^{n-1}$$ . In the case when $$n=2$$ this result is well known.

AGGREGATION AND IDEMPOTENCE

AbstractA 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this phenomenon, generalized to having arbitrary connectives playing the role of conjunction; among intermediate logics, LC, shows itself to occupy a crucial position in this regard, and to suggest a characterization, applicable to a broader range of consequence relations, in terms of a variant of the notion of idempotence we shall call componentiality. This is an analogue, for the consequence relations of propositional logic, of the notion of a conservative operation in universal algebra.

Gauss-Mean Value Formula for Triharmonic Functions and its Applications in Clifford Analysis

In this paper, the integral representation for some polyharmonic functions with values in a universal Clifford algebra Cl(Vn,n) is studied and Gauss-mean value formula for triharmonic functions with values in a Clifford algebra Cl(Vn,n) are proved by using Stokes formula and higher order Cauchy-Pompeiu formula. As application some results about growth condition at infinity are obtained.

Dual attachment pairs in categorically-algebraic topology

The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general setting for developing the attachment theory, motivated by the concept of (L,M)-fuzzy topological space of T. Kubiak and A. Sostak.

Berezin transforms on noncommutative varieties in polydomains

Let Q be a set of polynomials in noncommutative indeterminates Zi,j, i∈{1,…,k}, j∈{1,…,ni}. In this paper, we study noncommutative varietiesVQ(H):={X={Xi,j}∈D(H):g(X)=0 for all g∈Q}, where D(H) is a regular polydomain in B(H)n1+⋯+nk and B(H) is the algebra of bounded linear operators on a Hilbert space H. Under natural conditions on Q, we show that there is a universal modelS={Si,j} such that g(S)=0, g∈Q, acting on a subspace of a tensor product of full Fock spaces. We characterize the variety VQ(H) and its pure part in terms of the universal model and a class of completely positive linear maps. We obtain a characterization of those elements in VQ(H) which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non-coisometric elements. We study the universal model S, its joint invariant subspaces and the representations of the universal operator algebras it generates: the variety algebraA(VQ), the Hardy algebra F∞(VQ), and the C⁎-algebra C⁎(VQ). Using noncommutative Berezin transforms associated with each variety, we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains. This includes various commutative cases which are closely connected to the theory of holomorphic functions in several complex variables and algebraic geometry.

The syntactic concept lattice: Another algebraic theory of the context-free languages?

The syntactic concept lattice is a residuated lattice associated with a given formal language; it arises naturally as a generalization of the syntactic monoid in the analysis of the distributional structure of the language. In this article we define the syntactic concept lattice and present its basic properties, and its relationship to the universal automaton and the syntactic congruence; we consider several different equivalent definitions, as Galois connections, as maximal factorizations and finally using universal algebra to define it as an object that has a certain universal (terminal) property in the category of complete idempotent semirings that recognize a given language, applying techniques from automata theory to the theory of context-free grammars (CFGs). We conclude that grammars that are minimal, in a certain weak sense, will always have non-terminals that correspond to elements of this lattice, and claim that the syntactic concept lattice provides a natural system of categories for representing the denotations of CFGs.

The Universal Askey-Wilson Algebra and DAHA of Type (C1∨,C1)

Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\not=1$. The universal Askey-Wilson algebra $\Delta_q$ is the associative $\mathbb F$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $\Delta_q$. The universal DAHA $\hat H_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t^{\pm1}_i\rbrace_{i=0}^3$ and relations (i) $t_i t^{-1}_i=t^{-1}_i t_i=1$; (ii) $t_i+t^{-1}_i$ is central; (iii) $t_0t_1t_2t_3=q^{-1}$. We display an injection of $\mathbb F$-algebras $\psi:\Delta_q\to\hat H_q$ that sends $A\mapsto t_1t_0+(t_1t_0)^{-1}$, $B\mapsto t_3t_0+(t_3t_0)^{-1}$, $C\mapsto t_2t_0+(t_2t_0)^{-1}$. For the map $\psi$ we compute the image of the three central elements mentioned above. The algebra $\Delta_q$ has another central element of interest, called the Casimir element $\Omega$. We compute the image of $\Omega$ under $\psi$. We describe how the Artin braid group $B_3$ acts on $\Delta_q$ and $\hat H_q$ as a group of automorphisms. We show that $\psi$ commutes with these $B_3$ actions. Some related results are obtained.

Some applications of infinitary logical languages in universal algebra

Some examples are given of applications of an innite logical language in universal algebra, in the algebraic geometry of universal algebras, in the theory of implicit operations on algebras, in the Galois theory between automorphisms of universal algebras and its subalgebras of xed points, in the theory of Hamiltonian closure of subalgebras and other areas.

A note on representations of some affine vertex algebras of type D

In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to $D_{\ell}^{(1)}$ of levels $n-\ell+1$, for $n \in \Z_{>0}$. For $n=1$, we study the representation theory of the quotient vertex operator algebra modulo the ideal generated by that singular vector. In the case $\ell =4$, we show that the adjoint module is the unique irreducible ordinary module for simple vertex operator algebra $L_{D_{4}}(-2,0)$. We also show that the maximal ideal in associated universal affine vertex algebra is generated by three singular vectors.

Rational equivalence of algebras, its clone generalizations, and clone categoricity

The clones of functions on a set A are related to an arbitrary universal algebra \(\mathfrak{A} = \left\langle {A;\sigma } \right\rangle \) in various natural ways. The simplest and minimal of them is the clone \(Tr(\mathfrak{A})\) of termal functions of the algebra \(\mathfrak{A}\), that is, the closure of the collection of signature functions of these algebras with respect to the operator of superposition. The coincidence of similar clones (up to conjugation by the bijections of the underlying sets of the algebras) generates various equivalence relations on universal algebras, the first of which is the relation of rational equivalence introduced by Mal’tsev. This article deals with clone equivalences of this kind between algebras of arbitrary signatures.

A Note on the Distributivity of the Huq Commutator Over Finite Joins

We show that the Huq commutator distributes over finite joins, in any semi-abelian algebraically cartesian closed category. As a consequence we show that for semi-abelian varieties of universal algebras (more generally for semi-abelian categories with large directed colimits of subobjects preserved, for each object B, by the functor B× −), the distributivity of the Huq commutator over joins is equivalent to algebraic cartesian closedness.

Universal algebra for general aggregation theory: Many-valued propositional-attitude aggregators as MV-homomorphisms

This article continues Dietrich and List's (2010) work on propositional-attitude aggregation theory, which is a generalized unification of the judgement aggregation and probabilistic opinion-pooling literatures. We first propose an algebraic framework for an analysis of (many-valued) propositional-attitude aggregation problems. Then we shall show that systematic propositional-attitude aggregators can be viewed as homomorphisms—algebraically structure-preserving maps—in the category of Chang's (1958a, Trans. Am. Math. Soc., 88, 467–490) MV-algebras. (Proof idea: Systematic aggregators are induced by maps satisfying certain functional equations, which in turn can be verified to entail homomorphy identities.) Since the 2-element Boolean algebra as well as the real unit interval can be endowed with an MV-algebra structure, we obtain as natural corollaries two famous theorems: Arrow's theorem for judgement aggregation as well as McConway's (1981, J. Am. Stat. Assoc., 76, 410–414) characterization of linear opinion pools. Conceptually, this characterization of aggregators can be seen as justifying a certain structuralist interpretation of social choice. Technically and perhaps more importantly, it opens up a new methodology to social-choice theorists: the analysis of general aggregation problems by means of universal algebra.

Hyperrings and [formula omitted]-relations. A general approach

The study of the equivalence relations of particular multialgebras for which the factor multialgebras are universal algebras satisfying certain identities is a very important and intensively studied topic in multialgebra theory and not only. Our paper provides a general multialgebraic approach for the construction of all these relations and also some important and interesting properties concerning the construction of the corresponding factor universal algebra. One of the purposes of our approach is to improve some results concerning α⁎-relations of hyperrings and Krasner hyperrings and we will do this in the last sections of this paper.

The universal associative envelope of the anti-Jordan triple system of [formula omitted] matrices

We show that the universal associative enveloping algebra of the simple anti-Jordan triple system of all n×n matrices (n⩾2) over an algebraically closed field of characteristic 0 is finite-dimensional. We investigate the structure of the universal envelope and focus on the monomial basis, the structure constants, and the center. We explicitly determine the decomposition of the universal envelope into matrix algebras. We classify all finite-dimensional irreducible representations of the simple anti-Jordan triple system, and show that the universal envelope is semisimple. We also provide an example to show that the universal enveloping algebras of anti-Jordan triple systems are not necessary to be finite-dimensional.

Lattice-valued soft algebras

Motivated by the rapidly developing theory of lattice-valued (or, more generally, variety-based) topological systems, which takes its origin in the crisp concept of S. Vickers (introduced as a common framework for both topological spaces and their underlying algebraic structures--frames or locales), the paper initiates a deeper study of one of its incorporated mathematical machineries, i.e., the realm of soft sets of D. Molodtsov. More precisely, we start the theory of lattice-valued soft universal algebra, which is based in soft sets and lattice-valued algebras of A. Di Nola and G. Gerla. In particular, we provide a procedure for obtaining soft versions of algebraic structures and their homomorphisms, as well as basic tools for their investigation. The proposed machinery underlies many concepts of (lattice-valued) soft algebra, which are currently available in the literature, thereby enabling the respective researchers to avoid its reinvention in future.