# Order Theory/Preordered classes and poclasses

**Definition (preordered class)**:

A **preordered class** is a set together with a binary relation satisfying the following axioms:

- (reflexivity)
- (transitivity)

**Definition (poclass)**:

A **poclass** (shorthand for **partially ordered class**) is a preordered class such that the following additional axiom is satisfied:

- 3. (antisymmetry)

**Example (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 .

**Definition (order homomorphism)**:

Let and be preordered classes. An **order homomorphism** from to is a class function so that for all .

**Definition (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 .

**Definition (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

- .

**Definition (product order)**:

Let be a family of preordered classes. The **product order** on the cartesian product is the order given by

- .