Semigroups with the ideal retraction property.
type: Item
instance of: scholarly article
title: Semigroups with the ideal retraction property.
zbMATH Open document ID: 1040.20047
author: Karen D. Aucoin
author: Jill A. Dumesnil
author: John A. Hildebrant
published in: Semigroup Forum
publication date: 2003-08-31T00:00:00Z
review text: A semigroup \(S\) has the (ideal) retraction property (RP) if \(S\) has at least one proper ideal and if any ideal \(I\) of \(S\) is a retract of \(S\) (i.e., there exists a homomorphism \(f\colon S\to I\) whose restriction to \(I\) is the identity function). This property is studied for some classes of semigroups; for example: (1) A semigroup \(S\) with zero has RP if for each ideal \(I\) of \(S\) there is a dual ideal \(J\) in \(S\) (i.e., \(I\cap J=\{0\}\) and \(I\cup J=S\)). (2) A semigroup \(S\) with zero, for which each congruence is a Rees congruence, has RP if and only if for each ideal of \(S\) there exists a dual ideal in \(S\) (that is, the congruence lattice of \(S\) is complemented). (3) A cyclic semigroup \(S\) has RP if and only if some generator of \(S\) has index two. Examples are given which show that RP is neither preserved by subsemigroups nor by direct products. Some classes of semigroups are indicated, for which RP is preserved by homomorphisms. Remark: A construction of all semigroups having the retraction property (in particular, finite or commutative) was given by \textit{E. J. Tully} jun. [J. Aust. Math. Soc. 9, 239--245 (1969; Zbl 0185.04802)].
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 1972503
zbMATH Keywords: ideal retraction property
zbMATH Keywords: Rees congruences
zbMATH Keywords: congruence lattices
reviewed by: H. Mitsch
MaRDI profile type: Publication
full work available at URL: s002330010149
OpenAlex ID: W2064309633
DOI: 10.1007/S002330010149
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description: scientific article; zbMATH DE number 1972503