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 groupoid, 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 $$\left({T, \circ}\right)$$ - 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.