A generalized permutahedron

With respect to a fixed n-element ordered set P, the generalized permutahedron Perm(P) is the set of all ordered sets P boolean AND L, where L is any permutation of the elements of the underlying n-element set. Considered as a subset of the extension lattice of an n-element set, Perm(P) is cover-preserving. We apply this to deduce, for instance, that, in any finite ordered set P, there is a comparability whose removal will not increase the dimension, and there is a comparability whose addition to P will not increase its dimension. We establish further properties about the extension lattice which seem to be of independent interest, leading for example, to the characterization of those ordered sets P for which this generalized permutahedron is itself a lattice.