A Note on Representations of Inverse Semigroups

Abstract

It is known [l; 2] that every inverse semigroup 5 has a faithful representation as a semigroup of (1, l)-mappings of subsets of a set A into A. The set A may be taken as the set of elements of 5 and the (1, l)-mappings as mappings of principal left ideals of 5 onto principal left ideals of 5. If £ is the set of idempotents of 5 then there is also a representation of 5, not necessarily faithful, as a semigroup of (1, l)-mappings of subsets of E into E [2]. If e££ denote by Se the subsemigroup eSe of 5. In this note we give a representation of any inverse semigroup S as a semigroup of isomorphisms between the semigroups Se. The representation is faithful if (a more general condition is given below) the center of each maximal subgroup of 5 is trivial.We recall that an inverse semigroup [3] is a semigroup S in which for any aES the equations xax -x and axa = a have a unique common solution xES called the inverse of a and denoted by a~l [5; 6]. This implies that the idempotents of 5 commute and that to each aES there corresponds a pair of idempotents e, f such that aa~l = e, a~la =/, ea = a, af=a. The idempotents e, f are called respectively the left and right units of a. For any two elements a, bES, (ab)~1 = b~ia~1 (see [3]). Throughout what follows 5 will denote an inverse semigroup and E will denote its set of idempotents.If e£E then S, will denote the subsemigroup eSe of 5. Lemma 1. If e,fEE then Ser\S/ = Sef.Proof. By Lemma 1 of [4] and its left-right dual SeC\Sf = Sef and eSr\fS = efS. Hence since Se = eSr\Se and Sf=fSr\Sf, it follows that S,r\Sf = efSr\Sef=Sef.