E-Inversive Semigroups With the Identity ABC = AC

A semigroup S is called an E-inversive if for every aS there exists x in S Such that axE(s), where E(s) is the set of all idempotents of S, introduced by G.Thierrin. The concept of sub direct product of two E-inversive semigroups introduced by H. Mitsch by using the concept of sub homomorphism of inverse semigroups introduced by Mc Alisterand N.R.Reilly. The semidirect of two E-inversive semigroups introduced by F.Catino and M.M.Miccoli. In this paper we study some special identities in an E-inversive semigroup and we present preliminaries and basic concepts of E-inversive semigroups.