  A binarization of a bounded variable $x$ is a linear formulation with
variables $x$ and additional binary variables $y_1,\dots, y_k$, so that
integrality of $x$ is implied by the integrality of $y_1,\dots, y_k$. A binary
extended formulation of a polyhedron $P$ is obtained by adding to the original
description of $P$ binarizations of some of its variables. In the context of
mixed-integer programming, imposing integrality on 0/1 variables rather than on
general integer variables has interesting convergence properties and has been
studied both from the theoretical and from the practical point of view.
  We propose a notion of \emph{natural} binarizations and binary extended
formulations, encompassing all the ones studied in the literature. We give a
simple characterization of the vertices of such formulations, which allows us
to study their behavior with respect to sequential convexification. %0/1
disjunctions. In particular, given a binary extended formulation and % a
binarization $B$ of one of its variables $x$, we study a parameter that
measures the progress made towards ensuring the integrality of $x$ via
application of sequential convexification. We formulate this parameter, which
we call rank, as the solution of a set covering problem and express it exactly
for the classical binarizations from the literature.
