Homomorphism

Here's a Venn diagram depicting how the different categories of homomorphisms relate to one another:

Abbreviation	Description
Mono	Set of monomorphisms (injective)
Epi	Set of epimorphism (surjective)
Iso	Set of isomorphisms (bijective)
Auto	Set of automorphisms (bijective and endomorphic)

 

A homomorphism is a correspondence between set A (the domain) and set B (the codomain or range), so that each object in A determines a unique object in B and each object in B has an arrow/function/morphism pointing to it from A.

If operations, for example, addition and multiplication, are defined for A and B, it is required that they correspond. That is, a * b must correspond to f(a) * f(b).

Homomorphisms preserve correspondence

Correspondence must be as follows:

• Single-valued: The morphism must at least be a partial function
• Surjective: Each a in A has at least one f(a) in B

Homomorphism is a way to compare two groups for structural similarities. It's a function between two groups that preserve their structure...