# Order Theory/Preordered classes and poclasses

Definition (preordered class):

A preordered class is a set ${\displaystyle S}$ together with a binary relation ${\displaystyle \leq \subset S\times S}$ satisfying the following axioms:

1. ${\displaystyle \forall s\in S:s\leq s}$ (reflexivity)
2. ${\displaystyle \forall r,s,t\in S:(s\leq r\wedge r\leq t\Rightarrow s\leq t)}$ (transitivity)

Definition (poclass):

A poclass (shorthand for partially ordered class) is a preordered class ${\displaystyle (S,\leq )}$ such that the following additional axiom is satisfied:

3. ${\displaystyle \forall s,t\in S:s\leq t\wedge t\leq s\Rightarrow s=t}$ (antisymmetry)

Example (subsets of the power set are ordered by inclusion):

Let ${\displaystyle S}$ be any set, and let ${\displaystyle \sigma \subset {\mathcal {P}}(X)}$. Then the relation on ${\displaystyle \sigma }$ defined by

${\displaystyle S\leq T:\Leftrightarrow S\subseteq T}$

is an order on ${\displaystyle \sigma }$.

Definition (order homomorphism):

Let ${\displaystyle (S,\leq )}$ and ${\displaystyle (T,\preceq )}$ be preordered classes. An order homomorphism from ${\displaystyle (S,\leq )}$ to ${\displaystyle (T,\preceq )}$ is a class function ${\displaystyle f:S\to T}$ so that ${\displaystyle x\leq y\Rightarrow f(x)\preceq f(y)}$ for all ${\displaystyle x,y\in S}$.

Definition (isotonic class function):

Let ${\displaystyle S,T}$ be sets, and let ${\displaystyle \leq _{S}}$ be a preorder on ${\displaystyle S}$, and ${\displaystyle \leq _{T}}$ a preorder on ${\displaystyle T}$. A class function ${\displaystyle f:S\to T}$ is said to be isotonic with respect to ${\displaystyle \leq _{S}}$ and ${\displaystyle \leq _{T}}$ iff ${\displaystyle f}$ is an order homomorphism from ${\displaystyle (S,\leq _{S})}$ to ${\displaystyle (T,\leq _{T})}$.

Definition (antitonic class function):

Let ${\displaystyle S,T}$ be sets with preorders ${\displaystyle \leq _{S},\leq _{T}}$ respectively. Then an antitonic class function from ${\displaystyle S}$ to ${\displaystyle T}$ with respect to the partial orders ${\displaystyle \leq _{S}}$ and ${\displaystyle \leq _{T}}$ is a class function ${\displaystyle f:S\to T}$ such that

${\displaystyle x\leq _{S}y\Rightarrow f(y)\leq _{T}f(x)}$.

Definition (product order):

Let ${\displaystyle (S_{\alpha },\leq _{\alpha })_{\alpha \in A}}$ be a family of preordered classes. The product order on the cartesian product ${\displaystyle \prod _{\alpha \in A}S_{\alpha }}$ is the order given by

${\displaystyle (s_{\alpha })_{\alpha \in A}\leq (t_{\alpha })_{\alpha \in A}:\Leftrightarrow \forall \alpha \in A:s_{\alpha }\leq t_{\alpha }}$.