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 C \mathcal {C} of the variety of all completely simple semigroups. We find an isomorphic copy of L ( C ) \mathcal {L}(\mathcal {C}) as a subdirect product of the lattices L ( R B ) \mathcal {L}(\mathcal {R}\,\mathcal {B}) , L ( A G ) \mathcal {L}(\mathcal {A}\,\mathcal {G}) , and L ( G ) \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.