Test for Submonoid

Theorem
To show that $$\left({T, \circ}\right)$$ is a submonoid of a monoid $$\left({S, \circ}\right)$$, we need to show that:


 * 1) $$T \subseteq S$$;
 * 2) $$\left({T, \circ}\right)$$ is a groupoid (i.e. that it is closed);
 * 3) $$\left({T, \circ}\right)$$ has an identity.

Proof
From Subsemigroup Closure Test, (1) and (2) are sufficient to show that $$\left({T, \circ}\right)$$ is a subsemigroup of $$\left({S, \circ}\right)$$.

Demonstrating the presence of an identity is then sufficient to show that it is a monoid.