Strict ordering


Definition

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

\(\mathcal{R}\) is asymmetric:
\(\forall x,y \in S : x{\mathcal{R}}y \Rightarrow \neg(y{\mathcal{R}}x)\)
\(\mathcal{R}\) is transitive:
\(\forall x,y,z \in S : x{\mathcal{R}}y \wedge y{\mathcal{R}}z \Rightarrow x{\mathcal{R}}z\)

Proper supersets