Subsemigroup Closure Test

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


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


 * $(2): \quad \left({T, \circ}\right)$ is a magma, i.e. that $\left({T, \circ}\right)$ is closed.

Proof
From Restriction of Operation Associativity, if $\circ$ is associative on $\left({S, \circ}\right)$, then it will also be associative on $\left({T, \circ}\right)$.

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

So, once we have established that $T \subseteq S$, all we need to do is to check for $\left({T, \circ}\right)$ to be closed.