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

## Definition

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

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

### Ordered Semigroup Isomorphism

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

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

### Ordered 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 $\left({S, +, \circ, \preceq}\right)$ and $\left({T, \oplus, *, \preccurlyeq}\right)$ be ordered rings.

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

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

## 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.