Definition:Order of Structure

Definition
The order of an algebraic structure $\left({S, \circ}\right)$ is the cardinality of its underlying set, and is denoted $\left|{S}\right|$.

That is, the order of $\left({S, \circ}\right)$ is the number of elements in $S$.

If the set $S$ is infinite, then $\left({S, \circ}\right)$ is of infinite order, or is described as an infinite structure.

Otherwise it is of finite order, or is described as a finite structure.

This definition is mostly used in the study of group theory.

Notation
Some sources use $o \left({S}\right)$ for the order of $S$, but this has problems of ambiguity with other uses of $o \left({n}\right)$. (See O-notation.)