The structure of commutative congruence compact monoids (Q1611024)

A semigroup \(S\) is called congruence compact if every filter base of congruence classes of \(S\) has a non-empty intersection. \textit{S. Bulman-Fleming}, \textit{E. Hotzel}, and \textit{P. Normak} [Mathematika 46, No. 1, 205-224 (1999; Zbl 0966.20025)] showed that every congruence compact commutative monoid \(S\) is a semilattice of its subsemigroups \(S_e=\{a\in S\mid a^n=e\) for some \(n\in\mathbb{N}\}\) for each \(e\in E(S)\), and that the semigroups \(S_e\) need not be congruence compact, in general. In this paper necessary and sufficient conditions for \(S_e\) (\(e\in E(S)\)) to have this property are given (in terms of principal ideals of \(S_e\)). Furthermore, the problem of the converse is dealt with, i.e., when a commutative semigroup \(S\), which is a semilattice \(Y\) of congruence compact semigroups of type \(S_e\), is congruence compact. In particular, this happens if \(Y\) is finite. Finally, it is shown that a semilattice \(Y\) of right-congruence compact monoids is congruence compact if and only if \(Y\) is finite.