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