Polytopes and posets associated to preorders

Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author. Preorder polytopes are shown to be lattice polytopes which satisfy a certain duality relating their Ehrhart polynomials with the zeta polynomials of their posets of lattice points. A combinatorial interpretation of the normalized volume of a preorder polytope is proven, together with formulas for the Ehrhart polynomial and the $h^\ast$-polynomial, and a combinatorial interpretation of the latter is conjectured. Several conjectures and results on the lattice point enumeration of arbor polytopes are generalized to preorder polytopes, new conjectures are proposed and new interesting examples of preorder polytopes are studied.