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We define the Augmentation property for binary matrices with respect todifferent rank functions. A matrix $A$ has the Augmentation property for agiven rank function, if for any subset of column vectors $x_1,...,x_t$ for forwhich the rank of $A$ does not increase when augmented separately with each ofthe vectors $x_i$, $1\leq i \leq t$, it also holds that the rank does notincrease when augmenting $A$ with all vectors $x_1,...,x_t$ simultaneously.This property holds trivially for the usual linear rank over the reals, but aswe show, things change significantly when considering the binary and booleanrank of a matrix.  We prove a necessary and sufficient condition for this property to hold underthe binary and boolean rank of binary matrices. Namely, a matrix has theAugmentation property for these rank functions if and only if it has a uniquebase that spans all other bases of the matrix with respect to the given rankfunction. For the binary rank, we also present a concrete characterization of afamily of matrices that has the Augmentation property. This characterization isbased on the possible types of linear dependencies between rows of $V$, inoptimal binary decompositions of the matrix as $A=U\cdot V$.  Furthermore, we use the Augmentation property to construct simple families ofmatrices, for which there is a gap between their real and binary rank andbetween their real and boolean rank.

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