Basic properties of E-inversive semigroups

Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-)inversive if for every a is an element of S(a not equal 0) th ere exists x is an element of S such that ax(not equal 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a is an element of S(not equal 0) there exist x, y is an element of S such that ax = ya(not equal 0) is an idempotent, is dealt with also. Ideal exte nsions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0- )inversive semigroups satisfying different cancellativity conditions is elu cidated. 1991 AMS classification number: 20M10.