Definition:Ordered Semigroup Automorphism

Definition
Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.

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


 * $(1): \quad$ A semigroup automorphism, that is, a semigroup isomorphism from the semigroup $\struct {S, \circ}$ to itself


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

Also see

 * Definition:Ordered Structure Automorphism