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 $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ 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
- $(2): \quad \forall k: k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$
- $(3): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$.
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
- $(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} + \map \phi {\mathbf y}$
- $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$
Ordered Structure Monomorphism
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 $\struct {T, \preccurlyeq}$.
Also see
- Definition:Order Embedding, also known as an order monomorphism
- Results about monomorphisms in the context of abstract algebra can be found here.
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.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras