Inverse Semigroups whose Lattices of Full Inverse Subsemigroups are Modular

Abstract

An inverse semigroup is said to be modular if its lattice LF (S) of full inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each D-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. It is shown that there is exactly one bisimple modular inverse semigroup which is not a group and that is nondistributive.