  We study ordered matroids and generalized permutohedra from a Hopf theoretic
point of view. Our main object is a Hopf monoid in the vector species of
extended generalized permutahedra equipped with an order of the coordinates;
this monoid extends the Hopf monoid of generalized permutahedra studied by
Aguiar and Ardila. Our formula for the antipode is cancellation-free and
multiplicity-free, and is supported only on terms that are compatible with the
local geometry of the polyhedron. Our result is part of a larger program to
understand orderings on ground sets of simplicial complexes (for instance, on
shifted and matroid independence complexes). In this vein, we show that shifted
simplicial complexes and broken circuit complexes generate Hopf monoids that
are expected to exhibit similar behavior.
