### Hall exponents of Boolean matrices

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Let $A$ be a square $(0,1)$-matrix. Then $A$ is a Hall matrix provided it has a nonzero permanent. The Hall exponent of $A$ is the smallest positive integer $k$, if such exists, such that ${A}^{k}$ is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing $A$ as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).

The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order $n$ which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class ${\Omega}_{n}$ of doubly stochastic matrices (convex hull of $n\times n$ permutation matrices). An alternative description of this partial order is given. We define a class of special faces of ${\Omega}_{n}$ induced by permutation matrices,...

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