$(0,1)$–matrices, discrepancy, and preservers. II

Abstract

Let $m$ and $n$ be positive integers, and let $R=(r_1,\dots ,r_m)$ and $S=(s_1,\dots ,s_n)$ be non-negative integral vectors. Let $\mathcal{A}(R,S)$ be the set of all $m\times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be non-increasing, and let $F(R,S)$ be the $m\times n$ $(0,1)$-matrix where for each $i$, the $i^{\text{th}}$ row of $F(R,S)$ consists of $r_i$ 1’s followed by $n–r_i$ 0’s, called Ferrers matrices. The discrepancy of an $m\times n$ $(0,1)$-matrix $A$, $\text{disc}(A)$, is the number of positions in which $F(R,S)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular we characterize linear operators that preserve both the set of Ferrers matrices and the set of matrices of discrepancy one.