Total preorder


Definition

A binary relation \(\mathrel{R}\), on a set \(S\), is a total preorder if and only if:

\(\mathrel{R}\) is reflexive:
\(\forall x,y \in S : x{\mathrel{R}}x\)
\(\mathrel{R}\) is transitive:
\(\forall x,y,z \in S : x{\mathrel{R}}y \wedge y{\mathrel{R}}z \Rightarrow x{\mathrel{R}}z\)
\(\mathrel{R}\) is connex:
\(\forall x,y \in S : x{\mathrel{R}}y \vee y{\mathrel{R}}x\)