Single elements of matrix incidence algebras
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 \(x\in X\), \(y\in Y\).
scientific article; zbMATH DE number 1547892