Ordered semigroup

Abstract: A partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotonehomomorphism 9 of S onto a partially ordered group such that {x e S: xty < x9) has a greatestmember. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in whichthe imposed partial order extends the natural partial order on the idempotents. The main tool used isa local structure theorem which is introduced in Section 2. This local structure theorem applies tomany other contexts as well. 1980 Mathematics subject classification (Amer. Soc.): 2 Math.0 M 10. 1. Partially ordered regular semigroups It is well known that the idempotents of a regular semigroup 5 can be partiallyordered by settingeuf if and only if e = ef = fe.This partial order is called the natural partial order on the idempotents of S. Weshall say that a partially ordered semigrou naturallyp S is partially ordered if theimposed partial order < extends the natural partial order on the idempotents;that is, ife = ef = fe implies e < f.Note there is no assumption in the definition that e < / implie — fe. s e = efAs we shall see, the existence of a partial order, extending the natural partial