# Subsemigroup Closure Test

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## Theorem

To show that an algebraic structure $\struct {T, \circ}$ is a subsemigroup of a semigroup $\struct {S, \circ}$, we need to show only that:

- $(1): \quad T \subseteq S$

- $(2): \quad \circ$ is a closed operation in $T$.

## Proof

From Restriction of Associative Operation is Associative, if $\circ$ is associative on $\struct {S, \circ}$, then it will also be associative on $\struct {T, \circ}$.

Thus we do not need to check for associativity in $\struct {T, \circ}$, as that has been inherited from its extension $\struct {S, \circ}$.

So, once we have established that $T \subseteq S$, all we need to do is to check for $\circ$ to be a closed operation.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.1$. Subsets closed to an operation - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 32.1$ Identity element and inverses