An application of polycyclic monoids to rings
type: Item
instance of: scholarly article
title: An application of polycyclic monoids to rings
zbMATH Open document ID: 0891.20041
author: Peter Hines
author: Mark V. Lawson
published in: Semigroup Forum
publication date: 1998-07-21T00:00:00Z
review text: The polycyclic inverse monoid \(P_\alpha\), where \(\alpha\) is countable, is the inverse monoid generated by the set \(\{p_i\mid 1\leq i\leq\alpha\}\) subject to the relations \(p_ip_j^{-1}=\delta_{ij}\). If \(\alpha=1\), this is the classical bicyclic monoid. It is shown that \(P_2\) contains a copy of \(P_n\) for every finite \(n\geq 2\). If \(P_n\) is a submonoid of (the multiplicative semigroup of) a ring \(R\) with identity and the zero of \(P_2\) is that of \(R\), then \(e=\sum^n_{i=1}p_i^{-1}p_i\) is an idempotent and \(eRe\simeq M_n(R)\). Moreover, if \(n=2\) and \(e=1\) then \(R\simeq M_k(R)\) for every \(k\geq 2\). Certain analogues of these results are discussed for semigroups of matrix type \({\mathcal M}(S,I,I,Q)\) over a monoid \(S\) containing \(P_\alpha\), \(\alpha\geq 2\).
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1106586
zbMATH Keywords: polycyclic inverse monoids
zbMATH Keywords: relations
zbMATH Keywords: bicyclic monoid
zbMATH Keywords: multiplicative semigroups
zbMATH Keywords: idempotents
reviewed by: Jan Okniński
MaRDI profile type: Publication
full work available at URL: s00233-002-7010-6
OpenAlex ID: W2092354902
DOI: 10.1007/S00233-002-7010-6
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description: scientific article; zbMATH DE number 1106586