nLab order


This entry is about the concept in order theory. See at group order for the concept of the same name in group theory.


In the sense of order theory

An order on a set SS 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:

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.

In the sense of group theory

A mostly unrelated notion from group theory is order of a group, meaning the cardinality |G||G| of the underlying set of a group GG, especially when this is finite.

By extension, one speaks of the order of an element xGx \in G, as the order of the cyclic subgroup x\langle x\rangle generated by the element. For example, the order of a permutation πS n\pi \in S_n is the least integer 1kn1 \le k\le n such that π k=id\pi^k = id.

Sometimes one thinks of an infinite group as having order zero. The orders then have the natural order relation of divisibility?.

Other meanings

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.

Last revised on December 25, 2023 at 02:52:36. See the history of this page for a list of all contributions to it.