HomoMorphism

#REDIRECT Homomorphism

Homomorphism

:''This word should not be confused with homeomorphism.'' In abstract algebra, a homomorphism is a map (mathematics) 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 (algebra)), 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

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 == ''Main article: kernel (algebra)'' 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. group (mathematics) or ring (algebra)), 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 (ring theory)). == Related topics == * morphism * continuous function * homeomorphism * diffeomorphism Abstract algebra

Homomorphism

''OK, but what does "morphism" mean at all???'' ---------------------- Er, so homomorphisms are the ''only'' kind of morphisms? If not, the redirect was incorrect. --:LMS From Mathworld: "A general morphism is called a homomorphism" -- User:The Anome Mathworld is incorrect. Some people use "homomorphism" and "morphism" interchangeably, as we do here, others use "homomorphism" for "morphisms of algebraic structures" (as opposed to analytic or topological structures). In the latter terminology, a continuous map would be considered a morphism, but not a homomorphism. --AxelBoldt I stand corrected - I come at this from a computer science/category theory angle. Time to call out the specialists! -- User:The Anome I wrote the redirect on the grounds that the article implied that what it was referring to was the general notion from category theory. (It said "The notion of morphism is studied abstractly in category theory.") I wanted an article called "morphism" to link to from other things that I am writing. (But was it bad form to have put a link to the redirect in "morphism" from the article "homomorphism" itself?) Upon further consideration of the article, however, I realised that this was not true. The article is written in the language of concrete categories (those where objects are sets with some structure and morphisms are functions that preserve that structure), which isn't such a crime, but unfortunately many of the statements made don't apply to all concrete categories. In particular, the bit about the equivalence relation is very specific, from universal algebra. So I rewrote the article to specify that it applied only to universal algebra (adding some features and cleaning up some notation as well -- change back any notation change that you feel is horrid). Then I wrote another article under the name "morphism" that holds generally, in any category, and using the more general language of abstract categories instead of concrete ones. I hope that it's clear now what the scopes of the 2 articles are, and that the distinction between the titles of the articles can be maintained. I'm fairly new here, so let me know if I screwed stuff up. (Thank goodness for the logs!) -- user:Toby Bartels

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Words begining with Homomorphism:

HomoMorphism
Homomorphism
Homomorphism
Homomorphisms
Homomorphism_(graph_theory)