Bicyclic semigroup

About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.

On commutative semigroup algebras

Abstract: This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).

Monoids: theme and variations (functional pearl)

Abstract: The monoid is a humble algebraic structure, at first glance even downright boring. However, there's much more to monoids than meets the eye. Using examples taken from the diagrams vector graphics framework as a case study, I demonstrate the power and beauty of monoids for library design. The paper begins with an extremely simple model of diagrams and proceeds through a series of incremental variations, all related somehow to the central theme of monoids. Along the way, I illustrate the power of compositional semantics; why you should also pay attention to the monoid's even humbler cousin, the semigroup; monoid homomorphisms; and monoid actions.

One-parameter inverse semigroups

Abstract: This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences. 0. Let G be a subgroup of the multiplicative group of positive real numbers and let P denote the subsemigroup of G consisting of all x E G with x ? 1. Denote by Wp the class of all inverse semigroups H for which there isa homomorphism f: P -> H such that f(P) generates H (no proper inverse subsemigroup of H contains f(P)). We shall call such semigroups H one-parameter inverse semigroups and denote by W= UP Wp the class of all one-parameter inverse semigroups. The class W contains well-known semigroups. For example, each homomorphic image of a subgroup of R, the positive real numbers, is a member of W. Also the bicyclic semigroup B is a member of W, as is seen by noting that B is generated by a copy of the nonnegative integers. Indeed, if H is any elementary inverse semigroup, then H1 is generated by a homomorphic image of the nonnegative integers, and so is a one-parameter inverse semigroup. The main purpose of this paper is to describe all one-parameter inverse semigroups. In the process of doing this, we shall construct what we term the free oneparameter inverse semigroups Fp, one for each subgroup G of R and its associated semigroup P. The semigroup Fp is the only inverse semigroup (up to isomorphism) generated by a subsemigroup isomorphic with P which has the property that each homomorphism f: P -> S, an inverse semigroup, extends uniquely to a homomorphism f: Fp -> S. In particular, every H E Wp is a homomorphic image of Fp. We thus adopt the point of view that by describing Fp and the lattice of congruences of Fp for arbitrary P, we will have described all one-parameter inverse semigroups. We shall assume a certain familiarity with the algebraic theory of semigroups, particularly inverse semigroups. (See Clifford and Preston [1].) The existence and uniqueness of Fp is a consequence of a theorem due to McAlister [3, Theorem 33]. We were greatly aided in the actual description of Fp Presented to the Society, August 27, 1969; received by the editors June 10, 1969. AMS 1970 subject classifications. Primary 20M10; Secondary 20M05, 22A15.