Emil W. Kiss

On Tractability and Congruence Distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called J´onsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of J´onsson terms are tractable by showing that such languages have bounded width. Consequently, the class of instances of the constraint satisfaction problem arising from such a constraint language that fail to have solutions is definable in Datalog.

Minimal sets and varieties

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally nite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie, [6]. We generalize part of this result by proving that all locally nite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.

An Easy Way to Minimal Algebras

A finite algebra C is called minimal with respect to a pair δ<θ of its congruences if every unary polynomial f of C is either a permutation, or f(θ)⊆δ. It is the basic idea of tame congruence theory developed by Ralph McKenzie and David Hobby [7] to describe finite algebras via minimal algebras that sit inside them. As shown in [7] minimal algebras have a very restricted structure. This paper presents a new tool, the Twin Lemma, which makes it possible to give short proofs of some of these structure theorems. This part can be read as an alternative introduction to the theory. Our method yields new information in the type 1 case, and is especially useful in describing E-minimal algebras (that is, algebras that are minimal with respect to every prime congruence quotient). We complete their theory given in [7] by proving a structure theorem for the type 1 case. Finally we show that if an algebra is minimal with respect to two quotients, then the two types are the same, and if this type is 2, 3, or 4, then the bodies are also equal.

Abelian algebras and the hamiltonian property

Kiss, E.W. and M.A. Valeriote, Abelian algebras and the Hamiltonian property, Journal of Pure and Applied Algebra 87 (1993) 37-49. We show that a finite algebra A is Hamiltonian if the class HS(AA) consists of Abelian algebras. As a consequence, we conclude that a locally finite variety is Abelian if and only if it3 is Hamiltonian. Furthermore, it is proved that A generates an Abelian variety if and only if A” is Hamiltonian. An algebra is Hamiltonian if every nonempty subuniverse is a block of some congruence on the algebra and an algebra is Abelian if for every term t(.x, j), the implication t(x, 7) = t(x, Z)+ t(w, 7) = t(w, Z) holds. Thus, locally finite Abelian varieties have definable principal congruences, enjoy the congruence extension property, and satisfy the RS-conjecture.

Three remarks on the modular commutator

First a problem of Ralph McKenzie is answered by proving that in a finitely directly representable variety every directly indecomposable algebra must be finite. Then we show that there is no local proof of the fundamental theorem of Abelian algebras nor of H. P. Gumms permutability results. This part may also be of interest for those investigating non-modular Abelian algebras. Finally we provide a Gumm-type characterization of the situation when twonot necessarily comparable congruences centralize each other. In doing this, we introduce a four variable version of the difference term in every modular variety. A “two-terms condition” is also investigated.

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.

Frames and rings in congruence modular varieties

Abstract Frames in modular lattices generate rings by von Neumanns extension of the classical representation theorem for projective geometries. In this paper it is shown that frames of congruence relations of algebras in congruence modular varieties generate rings which are also “endomorphism rings” in some sense. To achieve this goal, homomorphisms between Abelian congruences and certain congruences on free algebras are investigated. As an application, we relate the congruence identities of a modular variety to its ring.

A construction of semimodular lattices

In this paper we prove that if ℒ is a finite lattice, and r is an integral valued function on ℒ satisfying some very natural conditions, then there exists a finite geometric (that is, semimodular and atomistic) lattice I containing ℒ as a sublattice such that r is the height function of ℒ restricted to ℒ. Moreover, we show that if, for all intervals [e, f] of ℒ, semimodular lattices I, of length at most r(f)-r(e) are given, then I can be chosen to contain I in its interval [e, f] as a cover preserving {0}-sublattice. As applications, we obtain results of R. P. Dilworth and D. T. Finkbeiner.

The set of types of a finitely generated variety

Abstract 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 .

On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the dichotomy conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded width. Consequently, the class of instances of the constraint satisfaction problem arising from such a constraint language that fail to have solutions is definable in Datalog