# 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 $\left({S, \circ, \preceq}\right)$ 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, that is, 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.

### 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 $\left({R, +, \circ, \preceq}\right)$ be an ordered ring.

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

$(1): \quad$ An ordered group automorphism from the ordered group $\left({R, +, \preceq}\right)$ to itself
$(2): \quad$ A semigroup automorphism from the semigroup $\left({R, \circ}\right)$ to itself.

## 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.