# Definition:Monomorphism (Abstract Algebra)

## Definition

A homomorphism which is an injection is descibed as monic, or called a monomorphism.

### Semigroup Monomorphism

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

Let $\phi: S \to T$ be a (semigroup) homomorphism.

Then $\phi$ is a semigroup monomorphism if and only if $\phi$ is an injection.

### Group Monomorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: G \to H$ be a (group) homomorphism.

Then $\phi$ is a group monomorphism if and only if $\phi$ is an injection.

### Ring Monomorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.

Let $\phi: R \to S$ be a (ring) homomorphism.

Then $\phi$ is a ring monomorphism if and only if $\phi$ is an injection.

### Field Monomorphism

Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields.

Let $\phi: F \to K$ be a (field) homomorphism.

Then $\phi$ is a field monomorphism if and only if $\phi$ is an injection.

### R-Algebraic Structure Monomorphism

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures.

Then $\phi: S \to T$ is an $R$-algebraic structure monomorphism if and only if:

$(1): \quad \phi$ is an injection
$(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$
$(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$.

That is, if and only if:

$(1): \quad \phi$ is an injection
$(2): \quad \phi$ is an $R$-algebraic structure homomorphism.

This definition continues to apply when $S$ and $T$ are modules, and also when they are vector spaces.

### Vector Space Monomorphism

Let $V$ and $W$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space monomorphism if and only if:

$(1): \quad \phi$ is an injection
$(2): \quad \forall \mathbf x, \mathbf y \in V: \phi \left({\mathbf x + \mathbf y}\right) = \phi \left({\mathbf x}\right) + \phi \left({\mathbf y}\right)$
$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \phi \left({\lambda \mathbf x}\right) = \lambda \phi \left({\mathbf x}\right)$

### Ordered Structure Monomorphism

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 $\left({T, \preccurlyeq}\right)$.

## 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.