Lower Semimodular Inverse Semigroups

Abstract

An inverse semigroup S is said to be lower semimodular if its lattice LF(S) has that property. Previous results of the second author have essentially reduced the study of such semigroups to the 0-simple ones and, as we show, of those only the simple ones that are not groups are of concern. We show that such an inverse semigroup S is lower semimodular if and only if it is combinatorial, its semilattice of idempotents is archimedean in S and the poset of idempotents of each D-class contains at most two mutually incomparable elements, each of which is maximal. As easy corollaries, we recover the descriptions of modular and distributive inverse semigroups. With each lattice ordered group G, Reilly associated an "l-bisimple" inverse semigroup. We show that the bisimple lower semimodular inverse semigroups with totally ordered idempotents are, up to isomorphism, precisely the l-bisimple inverse semigroups over subgroups of the additive group of real numbers. There is, up to isomorphism, only one bisimple, non-group, lower semimodular inverse semigroup that is not of this type.