Definition:Order of Structure

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Definition

The order of an algebraic structure $\struct {S, \circ}$ is the cardinality of its underlying set, and is denoted $\order S$.


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


Infinite Structure

Let the underlying set $S$ of $\struct {S, \circ}$ be infinite.

Then $\struct {S, \circ}$ is an infinite structure.


Finite Structure

Let the underlying set $S$ of $\struct {S, \circ}$ be finite.

Then $\struct {S, \circ}$ a finite structure.


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.


Notation

Some sources use $\map o S$ for the order of $S$, but this has problems of ambiguity with other uses of $\map o n$. (See little-o notation.)


Also see

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

  • Results about the order of a group can be found here.


Sources