Maximality properties of some subsemigroups of Baer-Levi semigroups
type: Item
instance of: scholarly article
title: Maximality properties of some subsemigroups of Baer-Levi semigroups
zbMATH Open document ID: 0834.20064
DOI: 10.1007/BF02573628
published in: Semigroup Forum
publication date: 1996-03-06T00:00:00Z
full work available at URL: https://eudml.org/doc/135408
review text: Let \(T\) be a subset of a semigroup \(H\). The idealizer, \(\text{Id }T\), of the subset \(T\) is defined by \(\text{Id }T = \{h \in S : hT \subseteq T\) and \(Th \subseteq T\}\). The author proves a general result which tells how to find certain maximal subsemigroups of \(H\). He takes a subsemigroup \(T\) and a subset \(P\) of \(H\) which satisfy certain conditions. For each \({\mathcal J}\)-class \(Q\) of \(T\) such that \(Q \cap P \neq \emptyset\), he takes the greatest left ideal of \(H\) which is disjoint from \(Q\) and from these, he chooses an ideal \(L\) which is maximal within the collection and further satisfies \(T \subseteq \text{Id }L\). He then shows that \(\text{Id }L\) is a maximal subsemigroup of \(H\) and \(L\) is the greatest left ideal of \(H\) which is contained in \(\text{Id }L\). This result is applied repeatedly throughout the paper to find maximal subsemigroups of the Baer-Levi semigroups \(BL_X (q)\). These are semigroups of injective transformations on an infinite set. Specifically, let \(X\) be an infinite set and denote by \(M_X\) the semigroup of all injective transformations on \(X\). Choose a cardinal number \(q\) such that \(\aleph_0 \leq q \leq |X|\). The Baer-Levi semigroup \(BL_X (q)\) is defined by \(BL_X(q) = \{\varphi \in M_X : |X \setminus X\varphi|= q\}\). Among other things, it is shown that \(BL_X(q)\) has \(2^{2^{|X|}}\) maximal right simple subsemigroups.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 804687
zbMATH Keywords: idealizer
zbMATH Keywords: maximal subsemigroups
zbMATH Keywords: left ideal
zbMATH Keywords: Baer-Levi semigroups
zbMATH Keywords: semigroups of injective transformations
zbMATH Keywords: maximal right simple subsemigroups
author: Eckehart Hotzel
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 804687