$(0,1)$-matrices, discrepancy and preservers

Beasley, LeRoy B.

Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 1123-1131.

Résumé

Let $m$ and $n$ be positive integers, and let $R = (r_1, \ldots , r_m)$ and $S = (s_1,\ldots , s_n)$ be nonnegative integral vectors. Let $A(R,S)$ 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 nonincreasing, and let $F(R)$ be the $m \times n$ $(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1's followed by $(n-r_i)$ 0's. Let $A\in A(R,S)$. The discrepancy of A, ${\rm disc}(A)$, is the number of positions in which $F(R)$ 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 show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when $m=n$, the transpose mapping.

Détail

DOI : 10.21136/CMJ.2019.0092-18

Classification : 05B20, 05C50, 15A04, 15A21, 15A86
Keywords: Ferrers matrix; row-dense matrix; discrepancy; linear preserver; strong linear preserver