Lower semimodular inverse semigroups. (Q2461621)

An inverse subsemigroup of an inverse semigroup \(S\) is called full if it contains all idempotents of \(S\). All such subsemigroups form a lattice \(\mathcal{LF}(S)\). An inverse semigroup \(S\) is called lower semimodular if the lattice \(\mathcal{LF}(S)\) is lower semimodular. Lower semimodular inverse semigroups are described, besides groups and groups with adjoint zero. In particular, a completely \(0\)-simple inverse semigroup \(S\) that is not a group with adjoint zero is lower semimodular if and only if \(S\) is combinatorial and has at most three distinct non-zero idempotents. In this case \(\mathcal{LF}(S)\) is modular. The cases of simple and bisimple inverse semigroups also are considered.