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Homomorphism - Formal definition

Homomorphism - Formal definition: Encyclopedia II - Homomorphism - Formal definition

A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i.e. properties like identity elements, inverse elements, and binary operations. 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 gene ...

Homomorphism: Encyclopedia II - Homomorphism - Formal definition

Homomorphism - Formal definition

A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i.e. properties like identity elements, inverse elements, and binary operations.

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 sets with a single binary operation defined on them (an algebraic structure known as a magma), a homomorphism is a map such that

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 spaces)
algebra homomorphism

The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a homomorphism is a map between two algebraic structures of the same type such that

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