\begin{definition}[Definition:Ordered Semigroup Automorphism]
Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.
An '''ordered semigroup automorphism''' from $\struct {S, \circ, \preceq}$ to itself is a mapping $\phi: S \to S$ that is both:
:$(1): \quad$ A semigroup automorphism, that is, a semigroup isomorphism from the semigroup $\struct {S, \circ}$ to itself
:$(2): \quad$ An order isomorphism from the ordered set $\struct {S, \preceq}$ to itself.
\end{definition}