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.