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 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.