Single elements of matrix incidence algebras

An element s of an algebra A is called a single element of A if asb = 0 and a,b epsilon A imply that as = 0 or sb = 0. Let n epsilon Z(+), let K be a field and let less than or equal to be a partial order on {1, 2,..., n}. Let A(n) (less than or equal to) be the matrix incidence algebra consisting of those n x n matrices A = (a(i),(j)) with entries in K, satisfying a(i, J) = 0 whenever i not less than or equal to j. An element S = (s(i, j)) of A(n) (less than or equal to) is a single element if and only if (i) ri not equal 0 and c(j) not equal 0 double right arrow s(i, j) not equal 0, (ii) i less than or equal to j(1) and i less than or equal to j(2) for some i double right arrow r(j1) and r(j2) are linearly dependent, (iii) i(1) less than or equal to j and i(2) less than or equal to j for some j double right arrow c(i1) and c(i2) are linearly dependent. Here r(i) and c(j) denote the ith row and the jth column of S, respectively. If \K\ greater than or equal to 3, the maximum rank of a single element of A(n) (less than or equal to) is the largest positive integer m for which there exist sets X, Y of minimal, respectively, maximal, elements with \X\ = \Y\ = m satisfying x less than or equal to y for every x epsilon X, y epsilon Y. (C) 2000 Elsevier Science Inc. All rights reserved. AMS classification: Primary 15A30; Secondary 05B20.