Definition:Ordered Semigroup Automorphism

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

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


 * $(1): \quad$ A semigroup automorphism, i.e. a semigroup isomorphism, from the semigroup $\left({S, \circ}\right)$ to itself


 * $(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to itself.

Also see

 * Definition:Ordered Structure Automorphism