The Augmentation Property of Binary Matrices for the Binary and Boolean Rank

We define the Augmentation property for binary matrices with respect to different rank functions. A matrix A has the Augmentation property for a given rank function, if for any subset of column vectors x1, ..., xt for for which the rank of A does not increase when augmented separately with each of the vectors xi, 1 ≤ i ≤ t, it also holds that the rank does not increase when augmenting A with all vectors x1, ..., xt simultaneously. This property holds trivially for the usual linear rank over the reals, but as we show, things change significantly when considering the binary and boolean rank of a matrix. We prove a necessary and sufficient condition for this property to hold under the binary and boolean rank of binary matrices. Namely, a matrix has the Augmentation property for these rank functions if and only if it has a unique base that spans all other bases of the matrix with respect to the given rank function. For the binary rank, we also present a concrete characterization of a family of matrices that has the Augmentation property. This characterization is based on the possible types of linear dependencies between rows of V , in optimal binary decompositions of the matrix as A = U · V . Furthermore, we use the Augmentation property to construct simple families of matrices, for which there is a gap between their real and binary rank and between their real and boolean rank.