Linear representations of semigroups of Boolean matrices

Abstract:

Let ${B_n}$ be the multiplicative semigroup of $n \times n$ matrices over the semiring 0, 1 under the operations “or” and “and". We show that the least possible degree of a faithful representation of ${B_n}$ over a field is ${2^n} - 1$ by studying representations of a subsemigroup of ${B_n}$. By different methods we answer the same question for the subsemigroups of Boolean matrices greater than or equal to some permutation matrix (Hall matrices) and greater than or equal to the identity (reflexive Boolean matrices). We prove every representation of the latter semigroup can be triangularized.