# 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:
$$\forall x,y \in S : x{\mathcal{R}}y \vee y{\mathcal{R}}x$$