# Definition:Ordered Group Automorphism

## Definition

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.

## 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.