# Definition:Ordered Field Isomorphism

## Definition

Let $\struct {S, +, \circ, \preceq}$ and $\struct {T, \oplus, *, \preccurlyeq}$ be ordered fields.

An ordered field isomorphism from $\struct {S, +, \circ, \preceq}$ to $\struct {T, \oplus, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both:

$(1): \quad$ An ordered group isomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$
$(2): \quad$ A group isomorphism from the group $\struct {S_{\ne 0}, \circ}$ to the semigroup $\struct {T_{\ne 0}, *}$

where $S_{\ne 0}$ and $T_{\ne 0}$ denote the sets $S$ and $T$ without the zeros of $S$ and $T$ respectively.

## Linguistic Note

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

Thus isomorphism means equal structure.