On ranks, Green classes, and the theory of determinants of Boolean matrices

V. B. Poplavskii

Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 42-60

Résumé

We consider the groupoid of all possible matrices over an arbitrary Boolean algebra with partial operation of matrix product. On this groupoid, we define the equivalence classes analogous to the Green classes $H,C,R,D,J$ for semigroups. We introduce the notion of the minor rank of a Boolean matrix. We show that the column, row, factorisation and minor ranks are invariants for the $J$-class of this groupoid, and the minor ranks do not exceed the column, row, factorisation and permanent ranks. The key result of this work explains the role of the Boolean determinant. We show that in some $J$-class there exists a square $n\times n$ matrix with nonzero determinant if and only if the column, row, factorisation and minor ranks of any matrix of this class are equal to each other and equal to $n$. All $n\times n$ matrices of this $J$-class have equal determinants, while the determinants of the square matrices of greater size are equal to zero.