Dual of Ordered Semigroup is Ordered Semigroup

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

Then its dual $\struct {S, \circ, \succcurlyeq}$ is also an ordered semigroup.

Proof
From Dual Ordering is Ordering, we have that $\struct {S, \succcurlyeq}$ is an ordered set.

We also note from the definition that $\struct {S, \circ}$ is a semigroup.

It remains to be demonstrated that $\succcurlyeq$ is compatible with $\circ$.

Recall that $\struct {S, \circ, \preccurlyeq}$ is an ordered semigroup.

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

Let $x, y \in S$ be arbitrary such that $x \succcurlyeq y$.

We have:

and similarly:

Hence the result.