Subsemigroup Closure Test

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Theorem

To show that an algebraic structure $\left({T, \circ}\right)$ 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