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In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure.

N.B. Some authors use the word homomorphism in a larger context than that of algebra. Some take it to mean any kind of structure preserving map (such as continuous maps in topology), or even a more abstract kind of map—what we term a morphism—used in category theory. This article only treats the algebraic context. For more general usage see the morphism article.

For example, if one considers a sets with a single binary operation defined on them (an algebraic structure known as a magma), a homomorphism is a map f : X → Y such that

f(u * v) = f(u) @ f(v)

where * is the operation on X and @ is the operation on Y.

Each type of algebraic structure has its own type of homomorphism. For specific definitions see:

• group homomorphism
• ring homomorphism
• module homomorphism
• linear operator (a homomorphism on vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (fors)
• algebra homomorphismA homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx) kF(x) F(x+y) F(x)+F(y) F(xy) F(x)F(y).

The notion of a homomorphism can be given a formal definition in the context of universal algebraUniversal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Basic idea From the point of view of universal algebra, an algebra is a set A together with a collection of operations on A. An n- ary operation on A, a field which studies ideas common to all algebraic structures. In this setting, a homomorphism φ : A → B is a map between two algebraic structures of the same type such that

φ(f_A(x₁, …, x_n)) = f_B(φ(x₁), …, φ(x_n))

for each n-ary operation f and for all x_i in A.

1 Types of homomorphisms

• An isomorphismIn mathematics, an isomorphism is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition: :The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part is a bijective homomorphism. Two objects are said to be isomorphic if there is an isomorphism between them. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned.

• An epimorphismIn the context of abstract algebra or universal algebra, an epimorphism is simply a surjective homomorphism. In the more general (and abstract) setting of category theory, an epimorphism (also called an epic morphism is a morphism f : X → Y such that is a surjective homomorphism.

• A monomorphism is an injective homomorphism.

• A homomorphism from an object to itself is called an endomorphism.

• An endomorphism which is also an isomorphism is called an automorphism.

The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the article on morphism for more details.

Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of universal algebra) this extra condition is automatically satisfied.