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