The structure of commutative congruence compact monoids

Abstract

An algebra A is called congruence compact if every filter base of congruence classes of A has a non-empty intersection. In ring and module theory and for abelian groups, this notion coincides with linear compactness for the discrete topology. The main aim of this note is to describe commutative congruence compact semigroups. Recall that although congruence compact commutative groups have been described by Leptin already in 1954 ([3]), there are only a few publications devoted to the study of congruence compact semigroups. In [1], congruence compact monoids for which Green’s relations J and H coincide are described; congruence compact acts over semigroups are discussed in [5]. Although the paper [1] gives necessary and sufficient conditions for a commutative monoid to be congruence compact, an important question has been left open. Namely, it has been shown in [1, Corollary 2.3] that every congruence compact commutative monoid S can be presented as a semilattice of its maximal one-idempotent subsemigroups Se , where Se = {s ∈ S: s = e for some n ∈ N} for each e ∈ E(S). It also has been shown [1, Example 5.1] that the subsemigroups Se of a congruence compact commutative semigroup S need not be congruence compact as semigroups. In this paper we will find necessary and sufficient conditions for subsemigroups Se of a commutative congruence compact semigroup to be congruence compact. As a starting point, let us recall the following (cf [1, Corollary 4.7]).