Definition:Ordered Structure Monomorphism

Definition

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

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

$(1): \quad$ A monomorphism, i.e. an injective homomorphism, from the structure $\struct {S, \circ}$ to the structure $\struct {T, *}$

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

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

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

$(1): \quad$ A group monomorphism from the group $\struct {S, \circ}$ to the group $\struct {T, *}$

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

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 $\struct {T, *}$.

Also see

• Definition:Monomorphism (Abstract Algebra)
• Definition:Order Embedding

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.