Infinite dual symmetric inverse monoids
type: Item
instance of: scholarly article
title: Infinite dual symmetric inverse monoids
zbMATH Open document ID: 1399.20075
author: James East
published in: Periodica Mathematica Hungarica
publication date: 2018-03-16T00:00:00Z
review text: Here, the dual symmetric inverse monoid \(\mathcal{I}_X^*\) on an infinite set \(X\) is studied, i.e. the semigroup of all bijections between quotients of \(X\). If \({\mathcal{S}_X}\) is the symmetric group, then the relative rank (Sierpinski rank) \({\text{rank}}(\mathcal{I}_X^*:{\mathcal{S}_X}) = 2\). From this, it follows that the semigroup \(\mathcal{I}_X^*\) can be generated by the symmetric group \({\mathcal{S}_X}\) together with two elements \(\alpha ,\beta \in \mathcal{I}_X^*\). The characterization of all such pairs \(\alpha ,\beta \) depends on whether the cardinal \(| X| \) is regular or singular. If \({\mathcal{E}_X}\) is the set of idempotents of \(\mathcal{I}_X^*\) and \(\mathcal{I}_X^*\) is generated by a set \({\mathcal{E}_X} \cup \Sigma \), \(\Sigma \subseteq \mathcal{I}_X^*\), then \(| \Sigma | \geqslant {2^{| X|}}\) and \(\mathcal{I}_X^*\) is generated by the set \(\Sigma \) alone. The semigroup \(\mathcal{I}_X^*\) also has the semigroup Bergman property, i.e. any countable subset of \(\mathcal{I}_X^*\) is contained in a 4-generator subsemigroup.
reviewed by: Jaak Henno
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 6851229
zbMATH Keywords: dual symmetric inversemonoids
zbMATH Keywords: symmetric groups
zbMATH Keywords: generators idempotents
zbMATH Keywords: semigroup Bergman property
zbMATH Keywords: Sierpinski rank
MaRDI profile type: Publication
full work available at URL: s10998-017-0194-z
OpenAlex ID: W2738241120
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DOI: 10.1007/S10998-017-0194-Z
description: scientific article; zbMATH DE number 6851229