Definition:Ordered Semigroup Automorphism

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

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


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


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