Abstract. Idempotent-consistent semigroups are defined by the property that each idempotent in a morphic image has an idempotent pre-image. We show that this class is strictly contained in the class of $E$-inversive semigroups while strictly containing a certain class of coextensions of eventually regular semigroups.

AMS Subject Classification (1991): 20M05, 20M17

Received October 28, 1998, and in final form July 12, 2000. (Registered under 2776/2009.)