nLab
order

An order on a set $S$ is (usually) a binary relation that is, at the very least, transitive.

Actually, there are several different notions of order that are each useful in their own ways:

• direction
• linear order
• partial order
• preorder
• quasiorder
• total order
• well-order

The closely related notion of a cyclic order is not actually a binary relation but a ternary relation.

The study of orders is order theory.