On a Class of Inverse Semigroups

Abstract

In this note we investigate the structure of a special class of inverse sernigroups-inverse semigroups the non-zero idempotents of which are primitive. We show that an inverse semigroup S has its non-zero idempoteits primitive if and only if it is a class sum of its Brandt ideals. A set of other equivalent conditions on S is also obtained. Brandt semigroups are treated by Mcfadden and Schneider [4] and Preston and Clifford [6]. The reader is referred to Bruck [1] and Preston and Clifford [6] for many results and literature on these subjects. A semigroup is a non-null multiplicative associative system. Throughout S denotes a semigroup with zero. A subset A of S is a left (right) ideal of S if SA CA (AS CA). A is a two-sided ideal of S if it is both a left and a right ideal of S. A left ideal B of S is minimal if B # (0) and if it contains no proper left ideal of S. Similarly minimal right (two-sided) ideal is defined. S is simple if it contains no proper two-sided ideal of S and 52 / (0). The principal left (right) ideal generated by an element a in S is a U Sa (a UaS). S is said to be regular if for any a in S there exists x in S such that a = axa. e = ax and f = xa are idempotents and ea = a af. e (f) is called a left (right) unit of a. S is regular if and only if for any a in 5, a U aS = eS (a U SaSf) for some idempotent e (fi) in S. A regular semigroup with commuting idempotents is inverse. An idempotent e in S is primitive if for any idempotent f in S, ef M fe = f implies f0 or fe. S is completely simple if S is simple and contains a non-zero primitive idempotent. S is a Brandt semigroup if: