Definition:Epimorphism (Abstract Algebra)
This page is about Epimorphism in the context of Abstract Algebra. For other uses, see Epimorphism.
Definition
A homomorphism which is a surjection is an epimorphism.
Semigroup Epimorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups.
Let $\phi: S \to T$ be a (semigroup) homomorphism.
Then $\phi$ is a semigroup epimorphism if and only if $\phi$ is a surjection.
Monoid Epimorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids.
Let $\phi: S \to T$ be a (monoid) homomorphism.
Then $\phi$ is a monoid epimorphism if and only if $\phi$ is a surjection.
Group Epimorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a (group) homomorphism.
Then $\phi$ is a group epimorphism if and only if $\phi$ is a surjection.
Ring Epimorphism
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.
Let $\phi: R \to S$ be a (ring) homomorphism.
Then $\phi$ is a ring epimorphism if and only if $\phi$ is a surjection.
Field Epimorphism
Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields.
Let $\phi: R \to S$ be a (field) homomorphism.
Then $\phi$ is a field epimorphism if and only if $\phi$ is a surjection.
$R$-Algebraic Structure Epimorphism
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 epimorphism if and only if:
- $(1): \quad \phi$ is a surjection
- $(2): \quad \forall k: k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$
- $(3): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$
This definition also applies to modules, and also to vector spaces.
Also known as
An epimorphism, in all contexts, can be described with the adjective epic.
Also see
- Definition:Homomorphism (Abstract Algebra)
- Definition:Monomorphism (Abstract Algebra)
- Definition:Isomorphism (Abstract Algebra)
- Results about epimorphisms in the context of abstract algebra can be found here.
Linguistic Note
The word epimorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix epi- meaning onto.
Thus epimorphism means onto (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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): epimorphism