# 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

## 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**.