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.

$\blacksquare$