On the Lattice of Congruences on Idempotent-Regular-Surjective Semigroups

Abstract

In this paper we consider the following two equivalence relations on the lattice 𝒞(S) of all congruences on a semigroup S with the set of idempotents E S : where kerρ = {x ∈ S: (x, x 2) ∈ ρ} and trρ = ρ ∩ (E S × E S ) (ρ ∈ 𝒞(S)). We show that if S is idempotent-surjective, then every κ-class ρκ (ρ ∈ 𝒞(S)) coincides with some interval [ρκ, ρκ]. We study idempotent pure and E-disjunctive congruences on an arbitrary idempotent-surjective semigroup. In particular, we show that if τ is the greatest idempotent pure congruence on an idempotent-surjective semigroup S, then S/τ is E-disjunctive (see Section 2). In Section 3 we investigate the trace class ρθ (ρ ∈ 𝒞(S)) of an arbitrary idempotent-surjective semigroup S. Also, we study the connections between ρ ∈ 𝒞(S) and four associated (with ρ) congruences ρκ, ρκ, ρθ, ρθ. In particular, using a conjunction between the least Clifford congruence ξ (on an eventually regular semigroup) and the congruences ξκ, ξθ, we give the necessary and sufficient conditions for ξ to be idempotent pure (see Section 4). In Section 5 we describe all E-unitary congruences on an arbitrary idempotent-surjective semigroup. Finally, some remarks concerning strongly E-reflexive congruences on an eventually regular semigroup are given (see Section 6).