# Definition:Order of Structure/Infinite Structure

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## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

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

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

That is, $\struct {S, \circ}$ is an **infinite structure** if and only if $\struct {S, \circ}$ is not a finite structure.

## Also known as

An **infinite structure** can also be described as an algebraic structure **of infinite order**.

## Also see

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem

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- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups