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