On equalizer-flat acts

S

: a right S-act A is strongly flat if and only if the functor A\otimes- (from the category of left S-acts into the category of sets) preserves both pullbacks and equalizers. Stenström gave two interpolation-type conditions whose conjunction describes strong flatness. In 1986, P. Normak studied these conditions separately, lablelling them (P) and (E), and investigated their relation to different types of flatness. Bulman-Fleming, in 1991, showed that pullback flatness and strong flatness actually coincide, and several papers have appeared in which condition (P) is discussed. To date, little work has been done on equalizer flat acts. This paper gives some new results connecting condition (E) and equalizer flatness, concentrating on situations in which the two coincide. A description is given of the completely simple and completely 0-simple semigroups (with 1 adjoined) over which this occurs.