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 $\left\vert{S}\right\vert$.

Thus, for a finite set $S$, the order of $\left({S, \circ}\right)$ is the number of elements in $S$.

Also defined as
Some sources do not define the order of a structure for an underlying set of infinite cardinality, restricting themselves to the finite case.

Also see
This definition is mostly used in the context of group theory:


 * Definition:Finite Group
 * Definition:Infinite Group

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 Little-O notation.)