Showing papers on "Bicyclic semigroup published in 1969"

On the closure of the bicyclic semigroup

Abstract: In ?I, two properties of T are established which hold for arbitrary S; namely, that B is a discrete open subspace of T and T\B is an ideal of T if it is nonvoid. In ?11, we introduce the notion of a topological inverse semigroup and establish several properties of such objects. Some questions are posed. In ?111, it is shown that if S is a topological inverse semigroup, then T\B is a group with a dense cyclic subgroup. ?IV contains a description of three examples of a topological semigroup which contains B as a dense proper subsemigroup. Finally, in ?V, we assume that S is a locally compact topological inverse semigroup and show that either B is closed in S or T is isomorphic with the last of the examples described in ?IV. A corollary about homomorphisms from B into a locally compact topological inverse semigroup is obtained which generalizes a result due to A. Weil [1, p. 96] concerning homomorphisms from the integers into a locally compact group. All spaces are topological Hausdorff in this paper. We state the definitions of Green's equivalence relations in a semigroup and the definition of an inverse semigroup. Green's relations S, 9, a and 9 on a semigroup S are defined by: agtb if and only if a u aS=b u bS, afb if and only if a u Sa=b u Sb, 4=S n 9 and 9=' o M. The notations Ra, La, Ha, and Da stand for the appropriate equivalence class of a in S. A semigroup S is an inverse semigroup provided each element x of S has a unique inverse; that is, an element x - 1 of S such that xx - lx = x and x - lxx - 1 = x - 1. For details about inverse semigroups and Green's relations, see [2]. We assume a certain familiarity with these notions.

Matrix Representations of Inverse Semigroups

Abstract: In his paper [1], W. D. Munn determines the irreducible matrix representations of an arbitrary inverse semigroup. Munn also gives a necessary and sufficient condition upon a 0-simple inverse semigroup for it to have a non-trivial matirx representation and for such semigroups gives a complete account of their representations. Munn's results rest upon the earlier work of Clifford [2] in which the representations of Brandt semigroups were determined. An alternative account of such representations was given by Munn in [3]. This earlier work is presented in Sections 5.2 and 5.4 of [4].

Idempotent-Separating Extensions Of Inverse Semigroups

Abstract: Extensions of semigroups have been studied from two points of view; ideal extensions and Schreier extension. In this paper another type of extension is considered for the class of inverse semigroups. The main result (Theorem 2) is stated in the form of the classical treatment of Schreier extensions (see e.g.[7]). The motivation for the definition of idempotentseparating extension comes primarily from G. B. Preston's concept of a normal set of subsets of a semigroup [6]. The characterization of such extensions is applied to give another description of bisimple inverse ω-semigroups, which were first described by N. R. Reilly [8]. The main tool used in the proof of Theorem 2 is Preston's characterization of congruences on an inverse semigroup [5]. For the standard terminology used, the reader is referred to [1].

On the lattice of congruences on a regular semigroup

Abstract: A result of Reilly and Scheiblich for inverse semigroups is proved true also for regular semigroups. For any regular semigroup S the relation θ is defined on the lattice, Λ(S), of congruences on S by: (ρ, τ) ∈ θ if ρ and τ induce the same partition of the idempotents of S. Then θ is a congruence on Λ(S), Λ(S)/θ is complete and the natural homomorphism of Λ(S) onto Λ(S)/θ is a complete lattice homomorphism.

Self-injective semigroup rings for finite inverse semigroups

Abstract: 1. An identity for R(S) and the main theorem. Let S be an inverse semigroup, i.e. a regular semigroup in which idempotents commute. Let E be the set of idempotents in S. Then E is a commutative idempotent subsemigroup of S and each principal left (right) ideal of S has a unique idempotent generator [1, Theorem 1.17, p. 28]. Then R(S) has an identity if R(E) has one. If Z denotes the ring of integers and Z(E) has an identity, then R(E) has an identity. Note that E has a zero if I El is finite.

Transitive semigroup actions

Abstract: Following Wallace [15], we define an act to be a continuous function ,u: S x X X such that (i) S is a topological semigroup, (ii) X is a topological space, and (iii) ,u(s, ,u(t, x)) =,(st, r) for all s, t E S and x E X. We call (S, X, ,) an action triple, X the state space of the act, and we say S acts on X. We assume all spaces are Hausdorff and write sx for ,u(s, x). S is said to act transitively if Sx= X for all x E X and effectively if sx = tx for all x E X implies that s = t. The first section of this paper deals with transitive actions and especially with the case where the semigroup is simple. We obtain as a corollary that if S is a compact coninected semigroup acting transitively and effectively on a space X that contains a cut point, then K, the minimal ideal of S, is a left zero semigroup and X is homeomorphic to K.. A C-set is a subset, Y, of X with the property that if M is any continuum contained in X with M rc Y# 0, then either Mc Y or Yc M. In the second section, we consider the position of C-sets in the state space and prove as a corollary that if S is a compact connected semigroup with identity acting effectively on the metric indecomposable continuum, X, such that SX= X, then S must be a group. The author wishes to thank Professor L. W. Anderson for his patient advice and criticism.

A non-lorentzian form of the principle of relativistic invariance

Abstract: A linear quasistochastic representation of the hyperbolic tangent semigroup is described which acts in dual space-time and which is a semigroup of relativistic endomorphisms Some of its properties are described, and its relationship with the Lorentz group is discussed This transformation semigroup permits a novel approach to space-time concepts and a discussion of the representations of the quasistochastic semigroup of relativistic endomorphisms and the associated invariants