Some examples of bands with two components.
type: Item
instance of: scholarly article
title: Some examples of bands with two components.
zbMATH Open document ID: 1215.20058
DOI: 10.5565/PUBLMAT_54210_11
author: Ferran Cedó
author: Elena Rodríguez-Jorge
published in: Publicacions Matemàtiques
publication date: 2010-08-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pm/1277731544
review text: The paper is motivated by the following open problem: find necessary and sufficient conditions on a band \(S\) (a semigroup consisting of idempotents) in order that \(S\) is a linear semigroup (that is, \(S\) embeds into the multiplicative monoid of \(n\times n\) matrices over a field \(K\), for some \(n\geq 1\)). It is known that every band \(S\) is of the form \(S=\bigcup_{\gamma\in\Gamma}S_\gamma\), a semilattice \(\Gamma\) of rectangular bands \(S_\gamma\) (called components of \(S\)). Moreover, if \(S\) is linear then the semilattice \(\Gamma\) must be finite and every rectangular band is linear. So it is natural to start with the case where \(|\Gamma|=2\). Some progress on the above problem has been made by \textit{F. Cedó} and the reviewer [in Publ. Mat., Barc. 53, No. 1, 119-140 (2009; Zbl 1178.20054)] and by the authors in the paper [in Commun. Algebra 38, No. 11, 4117-4129 (2010; Zbl 1216.20055)]. In the paper under review this work is continued. An example is constructed, showing that certain natural finiteness conditions on one-sided annihilator ideals and on one-sided annihilator congruences in \(S\) are not sufficient for \(S\) to be linear, and a new conjecture is proposed. Moreover, an example of a band \(S\) with two components \(E\), \(F\) satisfying \(EFE=F\) is constructed such that certain commutative algebra \(R\) canonically associated to \(S\), in such a way that the semigroup algebra \(K[S]\) embeds into the algebra of upper triangular matrices \(T_7(R)\), does not satisfy the ascending chain condition on annihilator ideals, whence \(R\) itself does not embed into a matrix ring over a field.
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 5770011
zbMATH Keywords: bands
zbMATH Keywords: idempotents
zbMATH Keywords: semigroup algebras
zbMATH Keywords: linear semigroups
zbMATH Keywords: triangular matrices
reviewed by: Jan Okniński
MaRDI profile type: Publication
OpenAlex ID: W1782327529
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description: scientific article; zbMATH DE number 5770011