Projectivities of completely simple semigroups (Q2755741)

Let \(S={\mathcal M}(G,I,\Lambda,P)\) be a completely simple semigroup over a group \(G\). The lattice \(\text{Subcs}(S)\) of all completely simple subsemigroups of the semigroup \(S\) is studied. It is proved in particular that this lattice satisfies a non-trivial identity if and only if the lattice of all subgroups of the group \(G\) satisfies a non-trivial identity, and that this lattice is a complemented lattice if and only if the semigroup \(S\) is generated by idempotents. Some necessary and sufficient conditions modulo groups are found for two completely simple semigroups \(S\) and \(T\) given by their Rees matrix representations to have isomorphic lattices \(\text{Subcs}(S)\) and \(\text{Subcs}(T)\).