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- This word should not be confused with homeomorphism.
- 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.
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 xi 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.
- A homomorphism from an object to itself is called an endomorphism.
- An endomorphism which is also an isomorphism is called an automorphism.
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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