  The application of binary matrices are numerous. Representing a matrix as a
mixture of a small collection of latent vectors via low-rank factorization is
often seen as an advantageous method to interpret and analyze data. In this
work, we examine the minimal rank factorizations of binary matrices using
standard arithmetic (real and nonnegative) and logical operations (Boolean and
$\mathbb{Z}_2$). We examine all the relationships between the different ranks,
and discuss when the factorizations are unique. In particular, we characterize
when a Boolean factorization $X = W \land H$ has a unique $W$, a unique $H$
(for a fixed $W$), and when both $W$ and $H$ are unique, given a rank
constraint.
