The structure of commutative congruence compact monoids
type: Item
instance of: scholarly article
title: The structure of commutative congruence compact monoids
zbMATH Open document ID: 1005.20044
published in: Semigroup Forum
publication date: 2003-02-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 1784754
zbMATH Keywords: congruence classes
zbMATH Keywords: congruence compact commutative monoids
zbMATH Keywords: semilattices
zbMATH Keywords: principal ideals
zbMATH Keywords: commutative semigroups
zbMATH Keywords: congruence compact semigroups
author: Peeter Normak
reviewed by: H. Mitsch
MaRDI profile type: Publication
full work available at URL: s002330010021
OpenAlex ID: W2038934330
DOI: 10.1007/S002330010021
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description: scientific article; zbMATH DE number 1784754