Definition:Order of Structure/Infinite Structure

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

• Definition:Finite Structure

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