Semimodularity and weak modularity in subalgebra lattices of completely simple semigroups (Q1070354)

We consider the variety of completely simple semigroups as algebras with an additional unary operation \(x\to x^{-1}\), where \(x^{-1}\) is the inverse of x in the subgroup \(H_ x\). Those completely simple semigroups with modular and distributive subalgebra lattices have previously been characterized by the author. Here we determine those completely simple semigroups whose subalgebra lattices are semimodular, M-symmetric, and weakly modular.