# Preorder

## Definition

Let $$(S, \mathrel{R}$$ be a relational structure. Then $$\mathrel{R}$$ is a preordering on $$S$$ if and only if:

$$\mathrel{R}$$ is reflexive on $$S$$:
$$\forall x,y \in S : x{\mathrel{R}}x$$
$$\mathrel{R}$$ is transitive on $$S$$:
$$\forall x,y,z \in S : x{\mathrel{R}}y \wedge y{\mathrel{R}}z \Rightarrow x{\mathrel{R}}z$$
THEOREM

An antisymmetric preorder is a partial order.

Proof unavailable

THEOREM

A symmetric preorder is an equivalence relation.

Proof unavailable

THEOREM

Every symmetric preordering is an equivalence relation.

Proof available