Homomorphism

Definition

A homomorphism is a structure-preserving map between two algebraic structures of the same type. The precise definition depends on the type of structure, but homomorphisms always preserve the operations and relations that define the structure.

Examples by Structure

Group Homomorphisms

A function $f:G\to H$ between groups $(G,\cdot )$ and $(H,\ast )$ is a homomorphism if:

$f(a\cdot b)=f(a)\ast f(b)$

for all $a,b\in G$.

Ring Homomorphisms

A function $f:R\to S$ between rings is a homomorphism if:

• $f(a+b)=f(a)+f(b)$
• $f(a\cdot b)=f(a)\cdot f(b)$
• $f(1_R)=1_S$ (if rings are unital)

Module Homomorphisms

A function $f:M\to N$ between modules over a ring $R$ is a homomorphism if it is a group homomorphism that respects scalar multiplication:

$f(r\cdot m)=r\cdot f(m)$

Graph Homomorphisms

A function $f:G\to H$ between graphs preserves adjacency: if vertices $u$ and $v$ are adjacent in $G$, then $f(u)$ and $f(v)$ are adjacent in $H$.

Properties

• The composition of homomorphisms is a homomorphism
• The identity map is always a homomorphism
• Homomorphisms preserve identity elements
• Homomorphisms preserve inverses

Special Types

• Isomorphism: A bijective homomorphism with a homomorphic inverse
• Endomorphism: A homomorphism from a structure to itself
• Automorphism: An isomorphism from a structure to itself
• Monomorphism: An injective homomorphism
• Epimorphism: A surjective homomorphism

In Category Theory

In Category Theory, morphisms generalize the notion of homomorphism. Each category specifies what “structure-preserving map” means for its objects.

Related Concepts

• Morphism: The categorical generalization
• Isomorphism: Bijective homomorphisms
• Kernel: Measures failure of injectivity
• Image: The range of a homomorphism
• Group: One type of structure with homomorphisms
• Ring: Another type of structure with homomorphisms
• Module: Homomorphisms between modules
• Graph: Homomorphisms between graphs