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.

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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:

• direction
• strict total order
• partial order
• preorder
• strict preorder
• 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.

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.