On sandwich sets and congruences on regular semigroups

Abstract

Let S be a regular semigroup and E(S) be the set of its idempotents. We call the sets S(e, f)f and eS(e, f) one-sided sandwich sets and characterize them abstractly where e, f ∈ E(S). For a, a′ ∈ S such that a = aa′a, a′ = a′aa′, we call S(a) = S(a′a, aa′) the sandwich set of a. We characterize regular semigroups S in which all S(e; f) (or all S(a)) are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order.For every a ∈ S, we also define E(a) as the set of all idempotets e such that, for any congruence ϱ on S, aϱa 2 implies that aϱe. We study the restrictions on S in order that S(a) or \(E(a) \cap D_{a^2 } \) be trivial. For \(\mathcal{F} \in \{ \mathcal{S}, \mathcal{E}\} \), we define \(\mathcal{F}\) on S by a \(\mathcal{F}\) b if \(F(a) \cap F(b) \ne \not 0\). We establish for which S are \(\mathcal{S}\) or \(\mathcal{E}\) congruences.