Subsemigroup of Ordered Semigroup is Ordered

Theorem
Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.

Let $\struct {T, \circ_T}$ be a subsemigroup of $\struct {S, \circ}$.

Then the ordered structure $\struct {T, \circ_T, \preceq_T}$ is also an ordered semigroup.

In the above:
 * $\circ_T$ denotes the operation induced on $T$ by $\circ$
 * $\preceq_T$ denotes the restriction of $\preceq$ to $T \times T$.

Proof
It is necessary to ascertain that $\struct {T, \circ {\restriction_T} }$ fulfils the ordered semigroup axioms:

In this context, we see that $\text {OS} 0$ and $\text {OS} 1$ are fulfilled a fortiori by dint of $\struct {T, \circ {\restriction_T} }$ being a subsemigroup of $\struct {S, \circ}$.

We have that $\struct {S, \circ, \preceq}$ is an ordered semigroup.

From Restriction of Ordering is Ordering, we have that $\preceq_T$ is an ordering.

Hence:

and:

Hence $\preceq_T$ fulfils Ordered Semigroup Axiom $\text {OS} 2$ on $\struct {T, \circ_T, \preceq_T}$.