Bisimple inverse semi-groups

Abstract

In [1] Clifford showed that the structure of any bisimple inverse semigroup with identity is uniquely determined by that of its right unit subsemigroup. The object of this paper is to show that the structure of any bisimple inverse semigroup with or without identity is determined by that of any of its a-classes. Let us define a right partial semigroup S to be a set S together with a partial binary operation satisfying the following condition: (A) if, for elements a, b, c of S, a(bc) is defined then so also is (ab)c defined and then a(bc) = (ab)c. We say that a right partial semigroup S is isomorphic with a right partial semigroup T if there exists a bijection b of S onto T such that ab is defined if and only if aq bq is defined and such that if ab is defined then (ab)k = aq bb. We define an RP-system (R, P) to be a right partial semigroup R together with a subsemigroup P of R such that: P(1) ab is defined if and only if a E P, for all a, b in R; P(2) R has a left identity contained in P; P(3) ac = bc implies that a = b for all a, b E P, c E R; P(4) foralla,be R,PanPb=Pcforsomece R. It then follows from P(1) and P(3) that any left identity of R contained in P is, in fact, a two-sided identity for P and so is unique. Now consider any a-class R of a bisimple inverse semigroup S. If we define the partial binary operation o in R by the rule that a o b = ab if and only if ab e R, then with respect to this operation R is a right partial semigroup with a subsemigroup P such that (R, P) is an RP-system (Theorem 1.4). Note 1. RP-systems can be obtained from systems having products more generally defined than stipulated in P(1). We then just ignore products which do not satisfy the condition P(1) (cf. Examples 1, 2 in ?6). Note 2. In particular, R could be a lattice ordered group and P the positive cone of R. We show that for any RP-system (R, P) there exists a bisimple inverse semigroup, which we denote by R-1 o R, some a-class of which is isomorphic with R. Conversely, for any a-class R of a bisimple inverse semigroup S there is a subsemigroup P of R such that (R, P) is an RP-system and R1 o R is isomorphic with S.