Universal Algebra Research Articles

Generalized integral representations in Clifford analysis§

Based on the solution of the Helmholtz equation in this article, we get the generalized Cauchy integral formula, the Cauchy–Pompeiu formula, the higher-order Cauchy–Pompeiu formula, and the generalized integral representations related with the Helmholtz operator for functions with values in a universal Clifford algebra C(Vn,n). We also give weak solutions of the inhomogeneous equations Lku = f and , k ≥ 1, where Lu=Du+uh and , D is the Dirac operator. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Countable Lawvere Theories and Computational Effects

Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.

MULTI-HYPERSUBSTITUTIONS AND COLORED SOLID VARIETIES

Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multi-hypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.

On the direct limit of a direct system of multialgebras

This paper deals with multialgebras. An important tool in the theory of multialgebras is the fundamental relation, which brings us into the class of the universal algebras. In this paper we will present the construction of the direct limit of a direct system of multialgebras, some basic properties of this construction, and we will see that the fundamental algebra of the direct limit of multialgebras is the direct limit of the fundamental algebras of the given multialgebras.

On the subsemilattices of first-order definable and openly first-order definable congruences of the congruence lattice of a universal algebra

We prove some representation theorems for lattices and their lower subsemilattices as the lattices of congruences and subsemilattices of first-order definable congruences of universal algebras.

Ideals and clots in universal algebra and in semi-abelian categories

We clarify the relationship between basic constructions of semi-abelian category theory and the theory of ideals and clots in universal algebra. To name a few results in this frame, which establish connections between hitherto separated subjects, 0-regularity in universal algebra corresponds to the requirement that regular epimorphisms are normal; we describe clots in categorical terms and show that ideals are images of clots under regular epimorphisms; we show that the relationship between internal precrossed modules and internal reflexive graphs extends the relationship between compatible reflexive binary relations and clots.

General Theory of the Commutator for Deductive Systems. Part I. Basic Facts

The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of deductive systems. In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper.

Group actions and invariants in algebras of generic matrices

We show that the fixed elements for the natural GL m -action on the universal division algebra UD ( m , n ) of m generic n × n -matrices form a division subalgebra of degree n, assuming n ⩾ 3 and 2 ⩽ m ⩽ n 2 − 2 . This allows us to describe the asymptotic behavior of the dimension of the space of SL m -invariant homogeneous central polynomials p ( X 1 , … , X m ) for n × n -matrices. Here the base field is assumed to be of characteristic zero.

A basis for the non-crossing partition lattice top homology

We find a basis for the top homology of the non-crossing partition lattice Tn. Though Tn is not a geometric lattice, we are able to adapt techniques of Bjorner (A. Bjorner, On the homology of geometric lattices. Algebra Universalis 14 (1982), no. 1, 107---128) to find a basis with Cn?1 elements that are in bijection with binary trees. Then we analyze the action of the dihedral group on this basis.

Combinatorial problems raised from 2-semilattices

The Constraint Satisfaction Problem (CSP) provides a general framework for many combinatorial problems. In [A.A. Bulatov, A.A. Krokhin, P.G. Jeavons, Classifying the complexity of constraints using finite algebras, SIAM J. Comput. 34 (3) (2005) 720–742; P.G. Jeavons, On the algebraic structure of combinatorial problems, Theoret. Comput. Sci. 200 (1998) 185–204] and then in [A.A. Bulatov, P.G. Jeavons, Algebraic structures in combinatorial problems, Technical Report MATH-AL-4-2001, Technische Universität Dresden, Dresden, Germany, 2001], a new approach to the study of the CSP has been developed which uses properties of universal algebras assigned to certain subclasses of the CSP such that the time complexity and other properties of subclasses can be derived from the properties of the assigned algebras. In this paper we briefly survey this approach, and then prove that problem classes corresponding to finite 2-semilattices, that is groupoids satisfying the identities x x = x , x y = y x , x ( x y ) = ( x x ) y , can be solved in polynomial time. Making use of this result we classify finite conservative groupoids, and 4-element algebras with minimal clone of term operations with respect to the complexity of the corresponding CSP-class.

Definability and Interpolation in Non-Classical Logics

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

On Magari's concept of general calculus: notes on the history of tarski's methodology of deductive sciences

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn S , called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (Łoś, Suszko, Słupecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic. Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section.

Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems

An extension of parts of the theory of partially ordered varieties and quasivarieties, as presented by Palasinska and Pigozzi in the framework of abstract algebraic logic, is developed in the more abstract framework of categorical abstract algebraic logic. Algebraic systems, as introduced in previous work by the author, play in this more abstract framework the role that universal algebras play in the more traditional treatment. The aim here is to build the generalized framework and to formulate and prove abstract versions of the ordered homomorphism theorems in this framework.

On a characterization of the lattice of subsystems of a transition system

It was first proved by Birkhoff and Frink, and the result now belongs to the folklore, that any algebraic lattice is up to isomorphism the lattice of subuniverses of a universal algebra. A study of subsystems of a transition system yields a new algebraic concept, that of a strongly algebraic lattice. We give here a representation theorem to the manner of Birkhoff and Frink of such lattices.

EXPLICIT CONSTRUCTION AND UNIQUENESS FOR UNIVERSAL OPERATOR ALGEBRAS OF DIRECTED GRAPHS

Given a directed graph, there exist a universal operator algebra and universal C*-algebra associated to the directed graph. In this paper we give intrinsic constructions for these objects. We also provide an explicit construction for the maximal C*-algebra of an operator algebra. We discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isomorphism invariant for the universal operator algebra of a directed graph. We show that the underlying undirected graph is a Banach algebra isomorphism invariant for the universal C*-algebra of a directed graph.

An Algebraic Approach to Knowledge Base Models Informational Equivalence

In this paper we study the notion of knowledge from the positions of universal algebra and algebraic logic. We consider first order knowledge which is based on first order logic. We define categories of knowledge and knowledge base models. These notions are defined for the fixed subject of knowledge. The key notion of informational equivalence of two knowledge base models is introduced. We use the idea of equivalence of categories in this definition. We prove that for finite models there is a clear way to determine whether the knowledge base models are informationally equivalent.

Spinors in the hyperbolic algebra

The three-dimensional universal complex Clifford algebra C¯3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.

Fuzzy Horn logic II

The paper presents generalizations of results on so-called Horn logic, well-known in universal algebra, to the setting of fuzzy logic. The theories we consider consist of formulas which are implications between identities (equations) with premises weighted by truth degrees. We adopt Pavelka style: theories are fuzzy sets of formulas and we consider degrees of provability of formulas from theories. Our basic structure of truth degrees is a complete residuated lattice. We derive a Pavelka-style completeness theorem (degree of provability equals degree of truth) from which we get some particular cases by imposing restrictions on the formulas under consideration. As a particular case, we obtain completeness of fuzzy equational logic.

The Cohomology of the Universal Steenrod Algebra

The mod 2 universal Steenrod algebra Q is a homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra introduced in [1]. In this paper we show that Q is Koszul. It follows by [7] that its cohomology, being purely diagonal, is isomorphic to a completion of Q itself with respect to a suitable chain of two-sided ideals.