Actually, there are several different notions of order that are each useful in their own ways:
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.
A mostly unrelated notion from group theory is the cardinality of the underlying set of a group , especially when this is finite. By extension, one speaks of the order of an element , as the order of the cyclic subgroup generated by the element. For example, the order of a permutation is the least integer such that .
The term ‘order’ can also be used fairly generically as a synonym of ‘degree’ or ‘rank’, as in first-order logic, the order of a differential equation, etc. Of course, these various orders form a well-order, so this is not entirely unrelated either.