On semigroups with lower semimodular lattice of subsemigroups

The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960’s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green’s relation J is trivial have this property, a description in such terms is natural. In the case of periodic semigroups a case that turns out to include all eventually regular semigroups the characterization becomes quite explicit and yields interesting consequences. In the general case, it remains an open question whether there exists a simple, but not completely simple, semigroup with this property. Any such semigroup must at least be idempotent-free and D-trivial.