A preorder / partial order / total order relation on a set \(X\)
Preorder: a binary relation on \(X\) that is reflexive and transitive.
A partial order (poset) has the additional constraint that \(x \leq y \land y \leq x \implies x=y\)
We can always get a partial order from a preorder by quotienting, so it’s not that special.
A total order has all elements comparable.