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| {{distinguish|holomorphism|homeomorphism}}
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| In [[abstract algebra]], a '''homomorphism''' is a [[morphism|structure-preserving]] [[map (mathematics)|map]] between two [[algebraic structure]]s (such as [[group (mathematics)|group]]s, [[ring (mathematics)|ring]]s, or [[vector space]]s). The word ''homomorphism'' comes from the [[ancient Greek language]]: ''[[wikt:ὁμός|ὁμός]] (homos)'' meaning "same" and ''[[wikt:μορφή|μορφή]] (morphe)'' meaning "shape". [[Isomorphism]]s, [[automorphism]]s, and [[endomorphism]]s are special types of homomorphisms.
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| == Definition and illustration ==
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| === Definition ===
| | I'm a 49 years old, [http://Www.Google.Co.uk/search?hl=en&gl=us&tbm=nws&q=married&gs_l=news married] and study at the high school ([http://Browse.Deviantart.com/?qh=§ion=&global=1&q=Modern+Languages Modern Languages] and Classics).<br>In my free time I try to learn Arabic. I have been twicethere and look forward to returning anytime soon. I love to read, preferably on my ebook reader. I like to watch Arrested Development and How I Met Your Mother as well as documentaries about anything scientific. I like Squash.<br><br>Also visit my web site [http://freshgamehacks.com/carsfast-lightning-hack-tool-cheats-android-ios/ Cars Fast as Lightning hack] |
| A homomorphism is a map that preserves selected structure between two [[algebraic structure]]s, with the structure to be preserved being given by the naming of the homomorphism.
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| Particular definitions of homomorphism include the following:
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| * A [[semigroup homomorphism]] is a map that preserves an [[associative]] [[binary operation]].
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| * A [[monoid homomorphism]] is a semigroup homomorphism that maps the identity element to the identity of the codomain.
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| * A [[group homomorphism]] is a homomorphism that preserves the group structure. It may equivalently be defined as a semigroup homomorphism between groups.
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| * A [[ring homomorphism]] is a homomorphism that preserves the ring structure. Whether the multiplicative identity is to be preserved depends upon the definition of ''ring'' in use.
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| * A [[linear map]] is a homomorphism that preserves the vector space structure, namely the abelian group structure and scalar multiplication. The scalar type must further be specified to specify the homomorphism, e.g. every '''R'''-linear map is a '''Z'''-linear map, but not vice versa.
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| * An [[algebra homomorphism]] is a homomorphism that preserves the [[algebra over a field|algebra]] structure.
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| * A [[functor]] is a homomorphism between two [[category (mathematics)|categories]].
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| Not all structure that an object possesses need be preserved by a homomorphism. For example, one may have a semigroup homomorphism between two monoids, and this will not be a monoid homomorphism if it does not map the identity of the [[domain of a function|domain]] to that of the [[codomain]].
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| For example, a group is an algebraic object consisting of a [[set (mathematics)|set]] together with a single binary operation, satisfying certain axioms. If {{nowrap|(''G'', ∗)}} and {{nowrap|(''H'', ∗′)}} are groups, a '''homomorphism''' from {{nowrap|(''G'', ∗)}} to {{nowrap|(''H'', ∗′)}} is a function {{nowrap|''f'' : (''G'', ∗) → (''H'', ∗′)}} such that {{nowrap|1=''f''(''g''<sub>1</sub> ∗ ''g''<sub>2</sub>) = ''f''(''g''<sub>1</sub>) ∗′ ''f''(''g''<sub>2</sub>)}} for all elements {{nowrap|''g''<sub>1</sub>, ''g''<sub>2</sub> ∈ ''G''}}.
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| Since inverses exist in ''G'' and ''H'', one can show that the identity of ''G'' maps to the identity of ''H'' and that inverses are preserved.
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| The algebraic structure to be preserved may include more than one operation, and a homomorphism is required to preserve each operation. For example, a ring has both addition and multiplication, and a homomorphism from the ring {{nowrap|(''R'', +, ∗, 0, 1)}} to the ring {{nowrap|(''R''′, +′, ∗′, 0′, 1′)}} is a function such that {{nowrap|1=''f''(''r'' + ''s'') = ''f''(''r'') +′ ''f''(''s'')}}, {{nowrap|1=''f''(''r'' ∗ ''s'') = ''f''(''r'') ∗′ ''f''(''s'')}} and {{nowrap|1=''f''(1) = 1′}} for any elements ''r'' and ''s'' of the domain ring. If rings are not required to be unital, the last condition is omitted. In addition, if defining structures of (e.g. 0 and additive inverses in the case of a ring) were not necessarily preserved by the above, preserving these would be added requirements.
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| 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 {{nowrap|''f'' : ''A'' → ''B''}} is a function between two algebraic structures of the same type such that
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| :''f''(μ<sub>''A''</sub>(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)) = μ<sub>''B''</sub>(f(''a''<sub>1</sub>), ..., f(''a''<sub>''n''</sub>))
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| for each ''n''-ary operation ''μ'' and for all elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> ∈ ''A''}}.
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| === Basic examples ===
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| [[File:Exponentiation as monoid homomorphism svg.svg|thumb|x200px|[[Monoid]] homomorphism ''f'' from the monoid {{nowrap|{{color|#008000|('''N''', +, 0)}}}} to the monoid {{nowrap|{{color|#800000|('''N''', ×, 1)}}}}, defined by {{nowrap|1=''f''(''x'') = 2<sup>''x''</sup>}}. It is injective, but not surjective.]]
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| The [[real number]]s are a [[ring (mathematics)|ring]], having both addition and multiplication. The set of all 2 × 2 [[matrix (mathematics)|matrices]] is also a ring, under [[matrix addition]] and [[matrix multiplication]]. If we define a function between these rings as follows:
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| :<math>f(r) = \begin{pmatrix}
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| r & 0 \\
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| 0 & r
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| \end{pmatrix}</math>
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| where ''r'' is a real number, then ''f'' is a homomorphism of rings, since ''f'' preserves both addition:
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| :<math>f(r+s) = \begin{pmatrix}
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| r+s & 0 \\
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| 0 & r+s
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| \end{pmatrix} = \begin{pmatrix}
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| r & 0 \\
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| 0 & r
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| \end{pmatrix} + \begin{pmatrix}
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| s & 0 \\
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| 0 & s
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| \end{pmatrix} = f(r) + f(s)</math>
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| and multiplication: | |
| :<math>f(rs) = \begin{pmatrix}
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| rs & 0 \\
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| 0 & rs
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| \end{pmatrix} = \begin{pmatrix}
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| r & 0 \\
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| 0 & r
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| \end{pmatrix} \begin{pmatrix}
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| s & 0 \\
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| 0 & s
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| \end{pmatrix} = f(r)\,f(s).</math>
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| For another example, the nonzero [[complex number]]s form a [[group (mathematics)|group]] under the operation of multiplication, as do the nonzero real numbers. (Zero must be excluded from both groups since it does not have a [[multiplicative inverse]], which is required for elements of a group.) Define a function ''f'' from the nonzero complex numbers to the nonzero real numbers by
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| :''f''(''z'') = |''z''|. | |
| That is, ''ƒ''(''z'') is the [[absolute value]] (or modulus) of the complex number ''z''. Then ''f'' is a homomorphism of groups, since it preserves multiplication:
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| :''f''(''z''<sub>1</sub> ''z''<sub>2</sub>) = |''z''<sub>1</sub> ''z''<sub>2</sub>| = |''z''<sub>1</sub>| |''z''<sub>2</sub>| = f(''z''<sub>1</sub>) f(''z''<sub>2</sub>).
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| Note that ''ƒ'' cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
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| :|''z''<sub>1</sub> + ''z''<sub>2</sub>| ≠ |''z''<sub>1</sub>| + |''z''<sub>2</sub>|.
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| As another example, the picture shows a [[monoid]] homomorphism ''f'' from the monoid {{nowrap|('''N''', +, 0)}} to the monoid {{nowrap|('''N''', ×, 1)}}. Due to the different names of corresponding operations, the structure preservation properties satisfied by ''f'' amount to {{nowrap|1=''f''(''x'' + ''y'') = ''f''(''x'') × ''f''(''y'')}} and {{nowrap|1=''f''(0) = 1}}.
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| ==Informal discussion==
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| Because abstract algebra studies [[Set (mathematics)|sets]] endowed with [[Operation (mathematics)|operations]] that generate interesting structure or properties on the set, [[function (mathematics)|function]]s which preserve the operations are especially important. These functions are known as ''homomorphisms''.
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| For example, consider the [[natural number]]s with addition as the operation. A function which preserves addition should have this property: {{nowrap|1=''f''(''a'' + ''b'') = ''f''(''a'') + ''f''(''b'')}}. For example, {{nowrap|1=''f''(''x'') = 3''x''}} is one such homomorphism, since {{nowrap|1=''f''(''a'' + ''b'') = 3(''a'' + ''b'') = 3''a'' + 3''b'' = ''f''(''a'') + ''f''(''b'')}}. Note that this homomorphism maps the natural numbers back into themselves.
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| Homomorphisms do not have to map between sets which have the same operations. For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication. A function which preserves operation should have this property: {{nowrap|1=''f''(''a'' + ''b'') = ''f''(''a'') · ''f''(''b'')}}, since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of [[exponent]]s, {{nowrap|1=''f''(''x'') = ''e''<sup>''x''</sup>}} satisfies this condition: {{nowrap|1=2 + 3 = 5}} translates into {{nowrap|1=''e''<sup>''2''</sup> · ''e''<sup>''3''</sup> = ''e''<sup>''5''</sup>}}.
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| If we are considering multiple operations on a set, then all operations must be preserved for a function to be considered as a homomorphism. Even though the set may be the same, the same function might be a group homomorphism, (a single binary operation, an inverse operation, being a unary operation, and identity, being a nullary operation) but not a ring isomorphism (two binary operations, the additive inverse and the identity elements), because it may fail to preserve the additional monoid structure required by the definition of a ring.
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| == Specific kinds of homomorphisms ==<!---renamed, since section is mainly not about category theory--->
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| [[Image:Morphisms3.svg|250px|thumb|Relationships between different kinds of homomorphisms. <br> ''Hom'' = set of Homomorphisms,<br> ''Mon'' = set of Monomorphisms, <br>''Epi'' = set of Epimorphisms,<br> ''Iso'' = set of Isomorphisms, <br>''End'' = set of Endomorphism,<br> ''Aut'' = set of Automorphisms.<br> Notice that: {{nowrap|1=''Mon'' ∩ ''Epi'' = ''Iso''}}, {{nowrap|1=''Iso'' ∩ ''End'' = ''Aut''}}. <br>The sets {{nowrap|(''Mon'' ∩ ''End'') \ ''Aut''}} and {{nowrap|(''Epi'' ∩ ''End'') \ ''Aut''}} contain only homomorphisms from some infinite structures to themselves.]]
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| {| style="float:right; border: 1px solid darkgray;"
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| |-
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| {| class="collapsible collapsed"
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| |-
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| ! colspan="3" | [Proof 1]
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| |-
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| | colspan="3" | For for each ''n''-ary operation ''μ'' and all ''b''<sub>1</sub>,...,''b''<sub>''n''</sub> ∈ ''B'':
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| |-
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| | || ''f''<sup>-1</sup>(μ<sub>''B''</sub>(''b''<sub>1</sub>,...,''b''<sub>''n''</sub>))
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| |-
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| | = || ''f''<sup>-1</sup>(μ<sub>''B''</sub>(''f''(''f''<sup>-1</sup>(''b''<sub>1</sub>)),...,''f''(''f''<sup>-1</sup>(''b''<sub>''n''</sub>))))
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| | since ''b'' = ''f''(''f''<sup>-1</sup>(''b'')) for each ''b'' ∈ ''B''
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| |-
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| | = || ''f''<sup>-1</sup>(''f''(μ<sub>''A''</sub>(''f''<sup>-1</sup>(''b''<sub>1</sub>),...,''f''<sup>-1</sup>(''b''<sub>''n''</sub>))))
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| | since ''f'' is a homomorphism
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| |-
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| | = || μ<sub>''A''</sub>(''f''<sup>-1</sup>(''b''<sub>1</sub>),...,''f''<sup>-1</sup>(''b''<sub>''n''</sub>))
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| | since ''a'' = ''f''<sup>-1</sup>(''f''(''a'')) for each ''a'' ∈ ''A''
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| |}
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| |-
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| {| class="collapsible collapsed"
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| |-
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| ! colspan="3" | [Proof 2]
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| |-
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| | colspan="3" | If ''g'' is a left inverse of ''f'',
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| |-
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| | colspan="3" | and ''f''(''g''<sub>1</sub>(''b'')) = ''f''(''g''<sub>2</sub>(''b'')), then
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| |-
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| | || ''g''<sub>1</sub>(''b'')
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| |-
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| | = || ''g''(''f''(''g''<sub>1</sub>(''b'')))
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| | since ''g'' is a left inverse of ''f''
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| |-
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| | = || ''g''(''f''(''g''<sub>2</sub>(''b'')))
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| | since ''f''(''g''<sub>1</sub>(''b'')) = ''f''(''g''<sub>2</sub>(''b''))
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| |-
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| | = || ''g''<sub>2</sub>(''b'')
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| | since ''g'' is a left inverse of ''f''
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| |}
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| |-
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| {| class="collapsible collapsed"
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| |-
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| ! colspan="3" | [Proof 3]
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| |-
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| | colspan="3" | If ''g'' is a right inverse of ''f'',
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| |-
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| | colspan="3" | and ''g''<sub>1</sub>(''f''(''a'')) = ''g''<sub>2</sub>(''f''(''a'')) for each ''a'' ∈ ''A'', then
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| |-
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| | || ''g''<sub>1</sub>(''b'')
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| |-
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| | = || ''g''<sub>1</sub>(''f''(''g''(''b'')))
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| | since ''g'' is a right inverse of ''f''
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| |-
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| | = || ''g''<sub>2</sub>(''f''(''g''(''b'')))
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| | since ''g''(''b'') ∈ ''A''
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| |-
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| | = || ''g''<sub>2</sub>(''b'')
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| | since ''g'' is a right inverse of ''f''
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| |}
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| |}
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| <!---sentences moved down: first define kinds, then discuss deviating notions in category theory below--->
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| <!---for indicating a contrast to category theory notions, the wording "modules and others" is confusing; I suggest "abstract algebra" instead---
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| In the important special case of [[module homomorphism]]s, and for some other classes of homomorphisms, there are much simpler descriptions, as follows:
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| --->
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| In abstract algebra, several specific kinds of homomorphisms are defined as follows:
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| * An '''[[isomorphism]]''' is a [[bijective]] homomorphism.
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| * An '''[[epimorphism]]''' (sometimes called a [[cover (algebra)|cover]]) is a [[surjective]] homomorphism. Equivalently, <ref group=note name="AC+nonconstr">tacitly assuming the [[axiom of choice]] and a [[constructive mathematics|nonconstructive setting]]</ref> ''f'': ''A'' → ''B'' is an epimorphism if it has a right inverse ''g'': ''B'' → ''A'', i.e. if ''f''(''g''(''b'')) = ''b'' for all ''b'' ∈ ''B''.
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| * A '''[[monomorphism]]''' (sometimes called an [[embedding]] or [[extension (model theory)|extension]]) is an [[injective]] homomorphism. Equivalently, <ref group=note name="AC+nonconstr"/> ''f'': ''A'' → ''B'' is a monomorphism if it has a left inverse ''g'': ''B'' → ''A'', i.e. if ''g''(''f''(''a'')) = ''a'' for all ''a'' ∈ ''A''.
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| * An '''[[endomorphism]]''' is a homomorphism from an algebraic structure to itself.
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| * An '''[[automorphism]]''' is an endomorphism which is also an isomorphism, i.e., an isomorphism from an algebraic structure to itself.
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| These descriptions may be used in order to derive several interesting properties. For instance, since a function is bijective if and only if it is both injective and surjective, in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.
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| An isomorphism always has an inverse ''f''<sup>−1</sup>, which is a homomorphism, too (cf. Proof 1).
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| If there is an isomorphism between two algebraic structures, they are completely indistinguishable as far as the structure in question is concerned; in this case, they are said to be ''isomorphic''.
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| ===Relation to category theory===
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| Since homomorphisms are [[morphism]]s, the above specific kinds of homomorphisms are [[Morphism#Some specific morphisms|specific kinds of morphisms]] defined in any category as well. However, the definitions in [[category theory]] are somewhat technical.
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| For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not.
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| In category theory, a morphism ''f'' : ''A'' → ''B'' is called:
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| * '''monomorphism''' if ''f'' ∘ ''g''<sub>1</sub> = ''f'' ∘ ''g''<sub>2</sub> implies ''g''<sub>1</sub> = ''g''<sub>2</sub> for all morphisms ''g''<sub>1</sub>, ''g''<sub>2</sub>: ''X'' → ''A'', where "∘" denotes function composition corresponding to e.g. (''f''∘''g''<sub>1</sub>)(''x'') = ''f''(''g''<sub>1</sub>(''x'')) in abstract algebra. (A sufficient condition for this is ''f'' having a left inverse, cf. Proof 2.)
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| * '''epimorphism''' if ''g''<sub>1</sub> ∘ ''f'' = ''g''<sub>2</sub> ∘ ''f'' implies ''g''<sub>1</sub> = ''g''<sub>2</sub> for all morphisms ''g''<sub>1</sub>, ''g''<sub>2</sub>: ''B'' → ''X''. (A sufficient condition for this is ''f'' having a right inverse, cf. Proof 3.)
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| * '''isomorphism''' if there exists a morphism ''g'': ''B'' → ''A'' such that ''f'' ∘ ''g'' = 1<sub>''B''</sub> and ''g'' ∘ ''f'' = 1<sub>''A''</sub>, where "1<sub>''X''</sub>" denotes the identity morphism on the object ''X''.<ref group=note>The notion of "object" and "morphism" in category theory generalizes the notion of "algebraic structure" and "homomorphism", respectively.</ref>
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| <!---deleted, since the notion of "bijective" doesn't appear in the "category theory" article---
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| For instance, the precise definition for a homomorphism ''f'' to be iso is not only that it is bijective, and thus has an inverse ''f''<sup>−1</sup>, but also that this inverse is a homomorphism, too.
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| --->
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| For instance, the inclusion of '''[[Integer|Z]]''' as a (unitary) subring of '''[[rational number|Q]]''' is not surjective (i.e. not epi in the abstract algebra sense), but an epimorphic [[ring homomorphism]] in the sense of category theory.<ref>Exercise 4 in section I.5, in [[Saunders Mac Lane]], ''[[Categories for the Working Mathematician]]'', ISBN 0-387-90036-5</ref> This inclusion thus also is an example of a ring homomorphism which is (in the sense of category theory) both mono and epi, but not iso.
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| == Kernel of a homomorphism ==
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| {{main|Kernel (algebra)}}
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| Any homomorphism {{nowrap|''f'' : ''X'' → ''Y''}} defines an [[equivalence relation]] ~ on ''X'' by {{nowrap|''a'' ~ ''b''}} if and only if {{nowrap|1=''f''(''a'') = ''f''(''b'')}}. The relation ~ is called the '''kernel''' of ''f''. It is a [[congruence relation]] on ''X''. The [[quotient set]] {{nowrap|''X'' / ~}} can then be given an object-structure in a natural way, i.e. {{nowrap|1=[''x''] ∗ [''y''] = [''x'' ∗ ''y'']}}. In that case the image of ''X'' in ''Y'' under the homomorphism ''f'' is necessarily [[isomorphic]] to {{nowrap|''X'' / ~}}; this fact is one of the [[isomorphism theorem]]s. Note in some cases (e.g. [[group (mathematics)|group]]s or [[ring (algebra)|ring]]s), a single [[equivalence class]] ''K'' suffices to specify the structure of the quotient; so we can write it ''X''/''K''. (''X''/''K'' is usually read as "''X'' [[Ideal (ring theory)|mod]] ''K''".) Also in these cases, it is ''K'', rather than ~, that is called the [[kernel (algebra)|kernel]] of ''f'' (cf. [[normal subgroup]]).
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| == Homomorphisms of relational structures ==
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| In [[model theory]], the notion of an algebraic structure is generalized to structures involving both operations and relations. Let ''L'' be a signature consisting of function and relation symbols, and ''A'', ''B'' be two ''L''-structures. Then a '''homomorphism''' from ''A'' to ''B'' is a mapping ''h'' from the domain of ''A'' to the domain of ''B'' such that
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| * ''h''(''F''<sup>''A''</sup>(''a''<sub>1</sub>,…,''a''<sub>''n''</sub>)) = ''F''<sup>''B''</sup>(''h''(''a''<sub>1</sub>),…,''h''(''a''<sub>''n''</sub>)) for each ''n''-ary function symbol ''F'' in ''L'',
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| * ''R''<sup>''A''</sup>(''a''<sub>1</sub>,…,''a''<sub>''n''</sub>) implies ''R''<sup>''B''</sup>(''h''(''a''<sub>1</sub>),…,''h''(''a''<sub>''n''</sub>)) for each ''n''-ary relation symbol ''R'' in ''L''.
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| In the special case with just one binary relation, we obtain the notion of a [[graph homomorphism]]. For a detailed discussion of relational homomorphisms and isomorphisms see.<ref>Section 17.4, in [[Gunther Schmidt]], 2010. ''Relational Mathematics''. Cambridge University Press, ISBN 978-0-521-76268-7</ref>
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| ==Homomorphisms and e-free homomorphisms in formal language theory==
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| Homomorphisms are also used in the study of [[formal language]]s<ref>[[Seymour Ginsburg]], ''Algebraic and automata theoretic properties of formal languages'', North-Holland, 1975, ISBN 0-7204-2506-9.</ref> (although within this context, often they are briefly referred to as morphisms<ref>T. Harju, J. Karhumӓki, Morphisms in ''Handbook of Formal Languages'', Volume I, edited by G. Rozenberg, A. Salomaa, Springer, 1997, ISBN 3-540-61486-9.</ref>). Given alphabets Σ<sub>1</sub> and Σ<sub>2</sub>, a function {{nowrap|''h'' : Σ<sub>1</sub><sup>∗</sup> → Σ<sub>2</sub><sup>∗</sup>}} such that {{nowrap|1=''h''(''uv'') = ''h''(''u'') ''h''(''v'')}} for all ''u'' and ''v'' in Σ<sub>1</sub><sup>∗</sup> is called a ''homomorphism'' (or simply ''morphism'') on Σ<sub>1</sub><sup>∗</sup>.<ref group=note>In homomorphisms on formal languages, the ∗ operation is the [[Kleene star]] operation. The ⋅ and ∘ are both [[concatenation]], commonly denoted by juxtaposition.</ref> Let ''e'' denote the empty word. If ''h'' is a homomorphism on Σ<sub>1</sub><sup>∗</sup> and {{nowrap|''h''(''x'') ≠ ''e''}} for all {{nowrap|''x'' ≠ ''e''}} in Σ<sub>1</sub><sup>∗</sup>, then ''h'' is called an ''e-free homomorphism''.
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| This type of homomorphism can be thought of as (and is equivalent to) a monoid homomorphism where Σ<sup>∗</sup> the set of all words over a finite alphabet Σ is a monoid (in fact it is the [[free monoid]] on Σ) with operation concatenation and the empty word as the identity.
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| ==See also==
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| * [[continuous function]]
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| * [[diffeomorphism]]
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| * [[homomorphic encryption]]
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| * [[homomorphic secret sharing]] – a simplistic decentralized voting protocol
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| * [[morphism]]
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| == Notes ==
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| {{reflist|group=note}}
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| ==References==
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| <div class="references-small">
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| {{refbegin}}
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| <references/>
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| A monograph available free online:
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| * Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2.
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| </div>
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| [[Category:Morphisms]]
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