# Definition:Isomorphism (Abstract Algebra)/Ordered Structure Isomorphism

## Definition

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

$(1): \quad$ An isomorphism, that is a bijective homomorphism, from the structure $\struct {S, \circ}$ to the structure $\struct {T, *}$
$(2): \quad$ An order isomorphism from the ordered set $\struct {S, \preceq}$ to the ordered set $\struct {T, \preccurlyeq}$.

### Ordered Semigroup Isomorphism

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

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

$(1): \quad$ A semigroup isomorphism from the semigroup $\struct {S, \circ}$ to the semigroup $\struct {T, *}$
$(2): \quad$ An order isomorphism from the ordered set $\struct {S, \preceq}$ to the ordered set $\struct {T, \preccurlyeq}$.

### Ordered Group Isomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered groups.

An ordered group isomorphism from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:

$(1): \quad$ A group isomorphism from the group $\left({S, \circ}\right)$ to the group $\left({T, *}\right)$
$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.

### Ordered Ring Isomorphism

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

An ordered ring 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 semigroup isomorphism from the semigroup $\struct {S, \circ}$ to the semigroup $\struct {T, *}$.

### Ordered Field Isomorphism

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.