Definition:Ordered Group Automorphism

Definition
Let $\left({S, \circ, \preceq}\right)$ be an ordered group.

An ordered group automorphism from $\left({S, \circ, \preceq}\right)$ to itself is a mapping $\phi: S \to S$ that is both:


 * A group automorphism, i.e. a group isomorphism, from the semigroup $\left({S, \circ}\right)$ to itself


 * An order isomorphism from the poset $\left({S, \preceq}\right)$ to itself.

Also see

 * Ordered Structure Automorphism