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