A binary relation \(\mathrel{R}\), on a set \(S\), is an ordering if and only if:

- \(\mathrel{R}\) is reflexive:
- \(\forall x \in S : x{\mathrel{R}}x\)
- \(\mathrel{R}\) 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 anti-symmetric:
- \(\forall x,y \in S : x{\mathrel{R}}y \wedge y{\mathrel{R}}x \Rightarrow x = y\)

```
order_refl: ?x ≤ ?x
order_trans: ?x ≤ ?y \(\Longrightarrow\) ?y ≤ ?z
order_antisym: ?x ≤ ?y \(\Longrightarrow\) ?y ≤ ?x \(\Longrightarrow\) ?x = ?y
```

A non-empty subset of an ordered set has one supremum at most.

A non-empty subset of an ordered set has one infimum at most.

An ordered set has one greatest element at most.

An ordered set has one smallest element at most.