Boolean-rank-preserving operators and Boolean-rank-1 spaces

Abstract

We study the extent to which certain theorems on linear operators on field-valued matrices carry over to linear operators on Boolean matrices. We obtain analogues and near analogues of several such theorems. One of these leads us to consider linear spaces of m × n Boolean matrices whose nonzero members all have Boolean rank 1. We obtain a structure theorem for such spaces that enables us to determine the maximum Boolean dimension of such spaces and their maximum cardinality.