## $$(0,1)$$-matrices, discrepancy and preservers.(English)Zbl 07144881

Summary: 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 $$A$$, $$\mathrm{disc}(A)$$, 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.

### MSC:

 15A04 Linear transformations, semilinear transformations 15A21 Canonical forms, reductions, classification 15A86 Linear preserver problems 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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### References:

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