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Homomorphism

This word should not be confused with homeomorphism.

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 sets with a single binary operation defined on them (an algebraic structure known as a magma), a homomorphism is a map $\phi: X \rightarrow Y$ such that

$\phi(u \cdot v) = \phi(u) \circ \phi(v)$

where $\cdot$ is the operation on $X$ and $\circ$ 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 $\phi: A \rightarrow B$ is a map between two algebraic structures of the same type such that

$\phi(f_A(x_1, \ldots, x_n)) = f_B(\phi(x_1), \ldots, \phi(x_n))$

for each n-ary operation $f$ and for all $x i$ in $A$ .

Types of homomorphisms

An isomorphism 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 epimorphism 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.

Kernel of a homomorphism

Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b). The relation ~ is called the kernel of f. It is a congruence relation on X. The quotient set X/~ can then be given an object-structure in a natural way, e.g., [x] * [y] = [x * y]. In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems. Note in some cases (e.g. groups or rings), a single equivalence class K suffices to specify the structure of the quotient, so we write it X/K. Also in these cases, it is K, rather than ~, that is called the kernel of f (cf. normal subgroup, ideal).

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