Test for Submonoid

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


 * $(1): \quad T \subseteq S$
 * $(2): \quad \struct {T, \circ}$ is a magma (that is, that it is closed)
 * $(3): \quad \struct {T, \circ}$ has an identity.

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

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