The congruence lattice of a regular semigroup

Abstract

Let S be a regular semigroup and Con S the congruence lattice of S. If C is an isomorphism class of semigroups and ϱϵ Con S, then we say that ϱ is over C if the idempotent ϱ-class belong to C . On Con S we can introduce the relations U, V, T l, T r and T as follows: if ϱ, θ, ϵ Con S, then we say that ϱ and θ are U− [ V−, T l−, T r−, T−] related if both ϱ/ϱ∩ θ and θ/ϱ∩ gq over completely simple semigroups [rectangular band, left groups, right groups, groups]. It is shown that U, V, T l, T r and T are complete congruences on Con S.Various other characterizations of these congruences on Con S are obtained. Some of the congruences are studied for completely regular semigroups, orthodox semigroups and bands of groups. Further, since for any regular semigroup S, V∩ T l∩ T r is the identity relation, we obtain a subdirect decomposition of Con S.