All Varieties of Central Completely Simple Semigroups

Abstract

Completely simple semigroups may be considered as a variety of algebras with the binary operation of multiplication and the unary operation of inversion. A completely simple semigroup is central if the product of any two idempotents lies in the centre of the containing maximal subgroup. Central completely simple semigroups form a subvariety $\mathcal {C}$ of the variety of all completely simple semigroups. We find an isomorphic copy of $\mathcal {L}(\mathcal {C})$ as a subdirect product of the lattices $\mathcal {L}(\mathcal {R} \mathcal {B})$, $\mathcal {L}(\mathcal {A} \mathcal {G})$, and $\mathcal {L}(\mathcal {G})$ of all varieties of rectangular bands, abelian groups, and groups, respectively. We consider also several homomorphisms and study congruences they induce.