On sandwich sets and congruences on regular semigroups
Mario Petrich · Czechoslovak Mathematical Journal · 2006
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 $$F(a) \cap F(b) e ot 0$$ . We establish for which S are $$\mathcal{S}$$ or $$\mathcal{E}$$ congruences.