On semigroups whose nontrlvial left congruence classes are left ideals

Abstract

An equivalence relation δ on a semigroup S is called a left congruence of S if (u, v) Ε λ implies that (su, sv) Ε λ for every s in S. With every set ℒ of pairwise disjoint left ideals (i. e. subsets L of S such that SL⊂EL), one can associate the left congruence {(u, v)|u = v or there exists an L in ℒ such that u Ε L and v Ε L}. Thus every nonempty left ideal is a left congruence class (i. e. an equivalence class of some left congruence). A left congruence has the form just described if and only if all its nontrivial classes (i. e. its classes containing at least two elements) are left ideals. Such a left congruence is called a Rees left congruence if there is at most one nontrivial class. The identity relation on S is a Rees left congruence since the empty set is a left ideal by definition.