# nLab order

Contents

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

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# Contents

## In the sense of order theory

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:

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 $|G|$ of the underlying set of a group $G$, especially when this is finite.

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

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 April 30, 2019 at 10:24:43. See the history of this page for a list of all contributions to it.