# 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:
$$\forall x,y,z \in S : x{\mathcal{R}}y \wedge y{\mathcal{R}}z \Rightarrow x{\mathcal{R}}z$$