Structure Of Finite Algebras Research Articles

The pasting constructions for effect algebras

Abstract The aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.

On the number of finite algebraic structures

We prove that every clone of operations on a finite set A , if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting R for some finitary relation R over A . It follows that for a fixed finite set A , the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.

On tame algebras of semiregular tubular type

We describe the structure of tame finite dimensional algebras over an algebraically closed field having strongly simply connected Galois coverings and the Auslander–Reiten quivers consisting of semiregular tubes.

Decidability of absorption in relational structures of bounded width

The absorption theory of Barto and Kozik has proven to be a very useful tool in the algebraic approach to the Constraint Satisfaction Problem and the structure of finite algebras in general. We address the following problem: Given a finite relational structure \({\mathbb{A}}\) and a subset \({B \subseteq A}\), is it decidable whether B is an absorbing subuniverse? We provide an affirmative answer in the case when \({\mathbb{A}}\) has bounded width (i.e., the algebra of polymorphisms of \({\mathbb{A}}\) generates a congruence meet semidistributive variety). As a by-product, we confirm that in this case the notion of Jonsson absorption coincides with the usual absorption. We also show that several open questions about absorption in relational structures can be reduced to digraphs.

Unary operations with long pre-periods

It is well known that the congruence lattice Con A of an algebra A is uniquely determined by the unary polynomial operations of A (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 [2]]). Let A be a finite algebra with | A | = n . If Im f = A or | Im f | = 1 for every unary polynomial operation f of A , then A is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 [3]]. If f : A → A is not a permutation then A ⊃ Im f and there is a least natural number λ ( f ) with Im f λ ( f ) = Im f λ ( f ) + 1 . We consider unary operations with λ ( f ) = n - 1 for n ⩾ 2 and λ ( f ) = n - 2 for n ⩾ 3 and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.

Nilpotent algebras and their applications

The work is devoted to the structure of finite nilpotent algebras and their applications, first of all, to the structure of products of groups. Some questions concerning the application of nilpotent algebras (and related structures) to the construction of recurrent sequences are touched upon.

AUTOMORPHISM GROUPS OF SQUARES AND OF FREE ALGEBRAS

We show that certain finite groups do not arise as the automorphism group of the square of a finite algebraic structure, nor as the automorphism group of a finite, 2-generated, free, algebraic structure.

Systems of Random Equations over Finite Algebraic Structures

The results on systems of random equations over finite algebraic structures are reviewed. Basic definitions, concepts, and problems in this field are presented.

Chief Factor Sizes in Finitely Generated Varieties

AbstractLet A be a k-element algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most ck−1. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to c in the situation where the variety generated by A omits type 1. As a generalization, we bound the size of multitraces of types 1, 2, and 3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

THE TYPE SET OF A VARIETY IS NOT COMPUTABLE

Several problems involving the local structure of finite algebras are shown to be unsolvable by interpreting the halting problem of Turing machines. Specifically, these problems are to decide, given a finite algebra A, whether the type-set of the variety generated by A, which is a subset of {1,2,3,4,5}, omits 2, or 3, or 4, or 5, respectively.

Foundations for Applications of Gibbs Derivatives in Logic Design and VLSI

New technologies and increased requirements for performances of digital systems require new mathematical theories and tools as a basis for future VLSI CAD systems. New or alternative mathematical approaches and concepts must be suitable to solve some concrete problems in VLSI and efficient algorithms for their efficient application should be provided. This paper is an attempt in this direction and relates with the recently renewed interest in arithmetic expressions for switching functions, instead representations in Boolean structures, and spectral techniques and differential operators in switching theory and applications.Logic derivatives are efficiently used in solving different tasks in logic design, as for example, fault detection, functional decomposition, detection of symmetries and co‐symmetries of logic functions, etc. Their application is based on the property that by differential operators, we can measure the rate of change of a logic function. However, by logic derivatives, we can hardly distinguish the direction of the change of the function, since they are defined in finite algebraic structures.Gibbs derivatives are a class of differential operators on groups, which applied to logic functions, permit to overcome this disadvantage of logic derivatives. Therefore, they may be useful in logic design in the same areas where the logic derivatives have been already using. For such applications, it is important to provide fast algorithms for calculation of Gibbs derivatives on finite groups efficiently in terms of space and time.In this paper, we discuss the methods for efficient calculation of Gibbs derivatives. These methods should represent a basis for further applications of these and related operators in VLSI CAD systems.

Complexity of Some Problems Concerning Varieties and Quasi-Varieties of Algebras

In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite algebras A and B, these problems ask the following. (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasi-identities? (3) Do A and B have the same set of term operations? In addition to the general case in which we allow arbitrary (finite) algebras, we consider each of these problems under the restrictions that all operations are unary and that A and B have cardinality two. We briefly discuss the relationship of these problems to algebraic specification theory.

Basic Hopf algebras and quantum groups

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers $\Gamma_G(W)$ given in terms of a finite group G and sequence $W=(w_1,w_2,\ldots,w_r)$ of elements of G in the following way can occur. The quiver $\Gamma_G(W)$ has vertices $\{v_g\}_{g\in G}$ and arrows $\{ (a_i,g)\colon v_{g^{-1}}\to v_{w_ig^{-1}}\mid g\in G, w_i\in W\}$ , where the set $\{ w_1,w_2,\ldots,w_r\}$ is closed under conjugation with elements in G and for each g in G, the sequences W and $(gw_1g^{-1}, gw_2g^{-1},\ldots, gw_rg^{-1})$ are the same up to a permutation. We show how $k\Gamma_G(W)$ is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras $k\Gamma_G(W)$ themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers $\Gamma_G(W)$ , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R].

Two finitely generated varieties having no infinite simple members

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote’s negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.

FREE SPECTRA GAPS AND TAME CONGRUENCE TYPES

Chapter 12 of "The Structure of Finite Algebras" by D. Hobby and R. McKenzie contains theorems revealing how the set of types appearing in a locally finite variety [Formula: see text] influences the size of the free algebra in [Formula: see text] freely generated by n elements. We provide more results in this vein. If A is a subdirectly irreducible algebra of size k, then a lower bound on the number of n-ary polynomials of A is obtained for each case that the monolith of A has type 3, 4, or 5. Examples for every k show that in each case the lower bound is the best possible. As an application of these results we show that for every finite k if all k-element simple algebras are partitioned into five classes according to their type, then algebras in each class have a sharply determined band of possible values for their free spectra. These five bands are disjoint except for some overlap on simple algebras of types 2 and 5.

On Solvable Congruences in Finitely Decidable Varieties

AbstractIn this paper we establish the (1, 2)‐ and (2, 1)‐transfer principles for finitely decidable locally finite varieties, where a class of structures is finitely decidable if the first order theory of its finite members is recursive. The transfer principles deal with the local structure of finite algebras and have strong global consequences.Mathematics Subject Classification: 08A05, 03C13, 08B05, 08B10.

The set of types of a finitely generated variety

The paper presents an algorithm of polynomial time complexity to compute the type set of a finite algebraic system A , as defined in the monograph of McKenzie and Hobby: ‘The Structure of Finite Algebras’. To do so, it introduces the concept of a subtrace, and uses subtraces to characterize the type set of A . It is also shown that to calculate the type set of the variety generated by A is more difficult, by presenting various examples, in which a given type occurs only in subalgebras of high powers of A .

Probabilistic properties of systems of random linear equations over finite algebraic structures

Probabilistic properties of systems of random linear equations over finite algebraic structures

Problems and results in tame congruence theory. A survey of the '88 Budapest workshop

Tame congruence theory is a powerful new tool, developed by Ralph McKenzie, to investigate finite algebraic structures. In the summer of 1988, many prominent researchers in this field visited Budapest, Hungary. This paper is a survey of problems and ideas that came up during these visits. It is intended both for beginners and experts, who want to do research, or just want to see what is going on, in this new, active area. An Appendix, written in April, 1990, is attached to the paper to summarize new developments.

On pseudovarieties, varieties of languages, filters of congruences, pseudoidentities and related topics

In recent years there has been considerable interest in pseudovarieties, mainly in the theory of finite semigroups and monoids. However, several results on pseudovarieties apply to much more general finite algebraic structures. This note provides a systematic approach to various characterizations of pseudovarieties, some old, some new, most of them formulated in terms of isomorphisms with the lattice of pseudovarieties contained in a given pseudovariety. It clarifies the relationships between alternative descriptions of pseudovarieties and it characterizes those sets of pseudoidentities which are(deductively) closed.