Kenneth McDowell

Left absolutely flat generalized inverse semigroups

A semigroup S is called (left, right) absolutely flat if all of its (left, right) S-sets are flat. S is a (left, right) generalized inverse semigroup if S is regula, and its set of idempotents E(S) is a (left, right) normal band (i.e. a strong semilattice of (left zero, right zero) rectangular bands). In this paper it is proved that a generalized inverse semigroup S is left absolutely flat if and only if S is a right generalized inverse semigroup and the (nonidentity) structure maps of E(S) are constant. In particular all inverse semigroups are left (and right) absolutely flat (see (1)). Other consequences are derived.

On V. Fleischer's characterization of absolutely flat monoids

In his paperCompletely flat monoids (Učh. Zap. Tartu Un-ta610 (1982), 38–52 (Russian)) V. Fleischer gives a characterization of the absolute flatness of a monoidS in terms of certain one-sided ideals and one-sided congruences ofS. In the present work an alternative, more direct proof of Fieischers theorem is provided, and the result is used to show that the multiplicative monoid of any semisimple Artinian ring is absolutely flat.

On left absolutely flat bands

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, s 6 T, a < s, and F is a finite subset of S3 x Ss, 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).

SOME OBSERVATIONS ON LEFT ABSOLUTELY FLAT MONOIDS

If S is any monoid a (unital) left S-set B is called fiat if the functor -| (from right S-sets into sets) preserves all embeddings of right S-sets, and weakly flat if this functor preserves embeddings of right ideals into S. S is called (weakly) left absolutely flat if all left S-sets are (weakly) flat. For a more complete discussion of these concepts consult [2] and the references cited therein. If S is a semigroup then OL(a,b) (resp. t3R(a,b)) will denote the smallest left (resp. right) congruence on S containing (a,b). Our first two observations concern weak left absolute flatness, and will make use of the following result: