Single elements of matrix incidence algebras Original Research Article

An element s of an algebra image is called a single element of image if asb=0 and image imply that as=0 or sb=0. Let image, let K be a field and let not precedes, equals be a partial order on {1,2,…,n}. Let image be the matrix incidence algebra consisting of those n×n matrices A=(ai,j) with entries in K, satisfying ai,j=0 whenever inot precedes, equals/j. An element S=(si,j) of image is a single element if and only if (i) ri≠0 and cj≠0impliessi,j≠0, (ii) inot precedes, equalsj1 and inot precedes, equalsj2 for some iimpliesrj1 and rj2 are linearly dependent, (iii) i1not precedes, equalsj and i2not precedes, equalsj for some jimpliesci1 and ci2 are linearly dependent. Here ri and cj denote the ith row and the jth column of S, respectively. If Kgreater-or-equal, slanted3, the maximum rank of a single element of image is the largest positive integer m for which there exist sets X,Y of minimal, respectively, maximal, elements with X=Y=m satisfying xnot precedes, equalsy for every xset membership, variantX, yset membership, variantY.