The polytope algebra of generalized permutahedra

The polytope subalgebra of deformations a zonotope can be endowed with the structure module over Tits algebra corresponding hyperplane arrangement. We explore this construction and find relations between statistics on (signed) permutations in case (type B) generalized permutahedra. In type B, surprisingly reveals that any family generators (via signed Minkowski sums) for permutahedra B will contain at least 2 d-1 full-dimensional polytopes. generating simplices attaining minimum. Finally, we prove defining are compatible Hopf monoid