The augmentation property of binary matrices for the binary and Boolean rank

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 sufficient 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.