Total preorder


Definition

Let \((S, \mathcal{R})\) be a relational structure. Then \(\mathcal{R}\) is a total preorder on \(S\) if and only if:

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

Proper supersets