CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES

Abstract

The Boolean rank of a nonzero m <TEX>$m{\times}n$</TEX> Boolean matrix A is the least integer k such that there are an <TEX>$m{\times}k$</TEX> Boolean matrix B and a <TEX>$k{\times}n$</TEX> Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with <TEX>$1{\leq}k{\leq}min\{m,n\}$</TEX>.