Subsemigroup Closure Test

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 Operation Associativity, 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.