Semidistributive Inverse Semigroups, II

The description by Johnston-Thom and the second author of the inverse semigroups S for which the lattice LJ(S) of full inverse subsemigroups of S is join semidistributive is used to describe those for which (a) the lattice L(S) of all inverse subsemigroups or (b) the lattice lo(S) of convex inverse subsemigroups have that property. In contrast with the methods used by the authors to investigate lower semimodularity, the methods are based on decompositions via GS, the union of the subgroups of the semigroup (which is necessarily cryptic)

Lower Semimodular Inverse Semigroups, II

The authors’ description of the inverse semigroups S for which the lattice ℒℱ(S) of full inverse subsemigroups is lower semimodular is used to describe those for which (a) the lattice ℒ(S) of all inverse subsemigroups or (b) the lattice �o(S) of convex inverse subsemigroups has that property. In each case, we show that this occurs if and only if the entire lattice is a subdirect product of ℒℱ(S) with ℒ(E S ), or �o(E S ), respectively, where E S is the semilattice of idempotents of S; a simple necessary and sufficient condition is found for each decomposition. For a semilattice E, ℒ(E) is in fact always lower semimodular, and �o(E) is lower semimodular if and only if E is a tree. The conjunction of these results leads to quite a divergence between the ultimate descriptions in the two cases, ℒ(S) and �o(S), with the latter being substantially richer