  We study the combinatorics of tropical hyperplane arrangements, and their
relationship to (classical) hyperplane face monoids. We show that the
refinement operation on the faces of a tropical hyperplane arrangement,
introduced by Ardila and Develin in their definition of a tropical oriented
matroid, induces an action of the hyperplane face monoid of the classical braid
arrangement on the arrangement, and hence on a number of interesting related
structures. Along the way, we introduce a new characterization of the types (in
the sense of Develin and Sturmfels) of points with respect to a tropical
hyperplane arrangement, in terms of partial bijections which attain permanents
of submatrices of a matrix which naturally encodes the arrangement.
