Positive matrix semigroups with binary diagonals

We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in S{\mathcal{S}} satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.