Single elements of matrix incidence algebras
type: Item
instance of: scholarly article
title: Single elements of matrix incidence algebras
zbMATH Open document ID: 0965.15015
DOI: 10.1016/S0024-3795(00)00165-8
author: Oreste Panaia
author: W. E. Longstaff
published in: Linear Algebra and its Applications
publication date: 2001-08-02T00:00:00Z
review text: An element \(s\) of an algebra \({\mathcal A}\) is called a single element of \({\mathcal A}\) if \(asb= 0\) and \(a,b\in{\mathcal A}\) imply that \(as= 0\) or \(sb= 0\). Let \(n\in \mathbb{Z}^+\), let \(K\) be a field and let \(\preceq\) be a partial order on \(\{1,2,\dots, n\}\). Let \({\mathcal A}_n(\preceq)\) be the matrix incidence algebra consisting of those \(n\times n\) matrices \(A= (a_{i,j})\) with entries in \(K\), satisfying \(a_{i,j}= 0\) whenever \(i \not\preceq j\). An element \(S= (s_{i,j})\) of \({\mathcal A}_n(\preceq)\) is a single element if and only if: (i) \({\mathbf r}_i\neq 0\) and \({\mathbf c}_j\neq 0\Rightarrow{\mathbf s}_{i,j}\neq 0\); (ii) \(i\preceq j_1\) and \(i\preceq j_2\) for some \(i\Rightarrow{\mathbf r}_{j_1}\) and \({\mathbf r}_{j_2}\) are linearly dependent; (iii) \(i_1\preceq j\) and \(i_2\preceq j\) for some \(j\Rightarrow c_{i_1}\) and \(c_{i_2}\) are linearly dependent. Here \({\mathbf r}_i\) and \({\mathbf c}_j\) denote the \(i\)th row and the \(j\)th column of \(S\), respectively. If \(|K|\geq 3\), the maximum rank of a single element of \({\mathcal A}_n(\preceq)\) is the largest positive integer \(m\) for which there exist sets \(X\), \(Y\) of minimal, respectively, maximal, elements with \(|X|=|Y|= m\) satisfying \(x\preceq y\) for every \(x\in X\), \(y\in Y\).
reviewed by: Rodica Covaci
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1547892
zbMATH Keywords: rank
zbMATH Keywords: minimal element
zbMATH Keywords: maximal element
zbMATH Keywords: single element
zbMATH Keywords: matrix incidence algebra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1547892