On left absolutely flat bands

Abstract

ABSTRACT. A semigroup S is called (left, right) absolutely flat if all of its (left, right) S-sets are flat. Let S = [_J{S~, : 7 6 T} be the least semilattice decomposition of a band S. It is known that if S is left absolutely flat then S is right regular (that is, each S7 is right zero). In this paper it is shown that, in addition, whenever a, ß 6 T, a < ß, and F is a finite subset of S3 x Sß, there exists w 6 Sa such that (wu,wv) € 6r{F) for all (u,v) € F (6r(F) denotes the smallest right congruence on S containing F). This condition in fact affords a characterization of left absolute flatness in certain classes of right regular bands (e.g. if T is a chain, if all chains contained in T have at most two elements, or if S is right normal).