Showing papers on "Bicyclic semigroup published in 1978"

Amenability of inverse semigroups and their semigroup algebras

Abstract: If G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.

The semigroup of varieties of Brouwerian semilattices

Abstract: It is shown that the semigroup of varieties of Brouwerian semilattices is free. Semigroups of varieties have first been studied in the case of groups, where Neumann, Neumann and Neumann-and independently Smerkin-have dis- covered a surprising result: The semigroup of varieties of groups is a free monoid with zero ((13), (14), (18)). Since then various authors have investi- gated semigroups of varieties of group-like structures such as quasigroups, rings, lattice-ordered groups and Lie-algebras ((2), (5), (8), (15), (19)). It was Mal'cev who considered the general case, and he succeeded in giving a sufficient condition under which the subvarieties of a given variety form a semigroup (7). iExploiting this idea Kohler studied the semigroup of varieties of Brouwerian algebras and Blok and Kohler did this for the semigroup of varieties of generalized interior algebras ((6), (1)). Nearly all the cited papers centered around the question whether a result similar to the one for groups could be obtained. This paper continues these efforts in giving a positive answer to the question above for the variety of Brouwerian semilattices. The paper is divided into two parts. The first one introduces the notion of an extension of a Brouwerian semilattice. Based on ideas originally in- troduced by Nemitz (10) and most elegantly generalized by Schmidt ((16), (17)) it is proven that every extension of a Brouwerian semilattice by another can be imbedded into some special kind of extension which we call strongly splitting extension. The second part introduces the multiplication of varieties of Brouwerian semilattices, thus giving the "set" of varieties a semigroup structure. Based on results from § 1 and using techniques originally developed in (6) it is finally proven that the semigroup of varieties is a free monoid with zero.

Lattice isomorphisms of inverse semigroups

Abstract: A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐx is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐx upon an inverse semigroup T is induced by a unique isomorphism of ℱℐx upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzuki's monograph on the subject of subgroup lattices (14)).

Subdirectly Irreducible Semigroups

Abstract: Definition 1.1. The ordered pair (S,*) is a semi-group iff S is a set and * is an associative binary operation (multiplication) on S. Notation. A semigroup (S,*) will ordinarily be referred to by the set S, with the multiplication understood. In other words, if (a,b)e SX , then *[(a,b)] = a*b = ab. The proof of the following proposition is found on p. 4 of Introduction to Semigroups, by Mario Petrich. Proposition 1.2. Every semigroup S satisfies the general associative law.

The left-discrimination sequence of an automaton

Abstract: Left-discrimination of a semigroup is defined and shown to be a sufficient condition that a semigroup be isomorphic to the input semigroup of its semigroup automaton, a necessary condition if the semigroup is finite. The left-discrimination sequence of an automaton is defined as a sequence of semigroups beginning with the input semigroup of the automaton, each member being the input semigroup of the semigroup automaton of its predecessor. It is related directly to a particular monotonically decreasing sequence of subautomata of the original automaton. This sequence is shown to be preserved by homomorphisms and is extended and used in an algorithm for determining the homomorphisms on one finite automaton to another.

Extensions de demi-groupes inverses

Abstract: We study the general problem of extension of one inverse semigroup by an another inverse semigroup. Any inverse semigroup can be rebuilt from its quotient by any congruence.