Order Theory/Preordered classes and poclasses

preordered class A preordered class is a set together with a binary relation satisfying the following axioms: (reflexivity) (transitivity) poclass A poclass (shorthand for partially ordered class) is a preordered class such that the following additional axiom is satisfied: 3. (antisymmetry) subsets of the power set are ordered by inclusion Let be any set, and let . Then the relation on defined by is an order on . order homomorphism Let and be preordered classes. An order homomorphism from to is a class function so that for all . isotonic class function Let be sets, and let be a preorder on , and a preorder on . A class function is said to be isotonic with respect to and iff is an order homomorphism from to . antitonic class function Let be sets with preorders respectively. Then an antitonic class function from to with respect to the partial orders and is a class function such that . product order Let be a family of preordered classes. The product order on the cartesian product is the order given by .