# Definition:Ordered Structure Monomorphism

## Definition

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

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

$(1): \quad$ A monomorphism, i.e. an injective homomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$
$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.

### Ordered Semigroup Monomorphism

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

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

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

### Ordered Group Monomorphism

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

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

### Ordered Ring Monomorphism

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

An ordered ring monomorphism 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 monomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$
$(2): \quad$ A semigroup monomorphism from the semigroup $\struct {S, \circ}$ to the semigroup $\struct {T, *}$.

## Linguistic Note

The word monomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix mono- meaning single.

Thus monomorphism means single (similar) structure.