Definition:Ordered Structure Automorphism

Definition

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

Let $\phi: S \to S$ be an ordered structure isomorphism from $S$ to itself.

Then $\phi$ is an ordered structure automorphism.

Ordered Semigroup Automorphism

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.

Ordered Group Automorphism

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

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

$(1): \quad$ A group automorphism, that is, a group isomorphism from the group $\left({G, \circ}\right)$ to itself

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

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.

Also see

• Definition:Automorphism (Abstract Algebra)

Linguistic Note

The word automorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus automorphism means self structure.