Structure Induced on Set of All Mappings to Ordered Semigroup is Ordered Semigroup

Theorem
Let $S$ be a set.

Let $\struct {T, \odot, \preccurlyeq}$ be an ordered semigroup.

Let $\struct {T^S, \otimes, \preccurlyeq}$ denote the algebraic structure on $T^S$ induced by $\odot$.

Then $\struct {T^S, \otimes, \preccurlyeq}$ is an ordered semigroup.

Proof
From Structure Induced by Semigroup Operation is Semigroup, $\struct {T^S, \otimes}$ is a semigroup

From Ordered Set of All Mappings is Ordered Set, $\struct {T^S, \preccurlyeq}$ is an ordered set.

It remains to be demonstrated that $\preccurlyeq$ is compatible with $\odot$.

Let $f, g \in T^S$ such that $f \preccurlyeq g$.

That is:
 * $\forall x \in S: \map f x \preccurlyeq \map g x$

We are given that $\struct {T, \odot, \preccurlyeq}$ be an ordered semigroup.

Hence $\preccurlyeq$ is compatible with $\odot$.

Thus:

Hence the result.