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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.
 
== Definition and illustration ==
 
=== Definition ===
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.
 
Particular definitions of homomorphism include the following:
* A [[semigroup homomorphism]] is a map that preserves an [[associative]] [[binary operation]].
* A [[monoid homomorphism]] is a semigroup homomorphism that maps the identity element to the identity of the codomain.
* A [[group homomorphism]] is a homomorphism that preserves the group structure. It may equivalently be defined as a semigroup homomorphism between groups.
* 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.
* 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.
* An [[algebra homomorphism]] is a homomorphism that preserves the [[algebra over a field|algebra]] structure.
* A [[functor]] is a homomorphism between two [[category (mathematics)|categories]].
 
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]].
 
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''}}.
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.
 
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.
 
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
:''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>))
for each ''n''-ary operation ''μ'' and for all elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> ∈ ''A''}}.
 
=== Basic examples ===
 
[[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.]]
The [[real number]]s are a [[ring (mathematics)|ring]], having both addition and multiplication.  The set of all 2&nbsp;×&nbsp;2&nbsp;[[matrix (mathematics)|matrices]] is also a ring, under [[matrix addition]] and [[matrix multiplication]].  If we define a function between these rings as follows:
:<math>f(r) = \begin{pmatrix}
  r & 0 \\
  0 & r
\end{pmatrix}</math>
where ''r'' is a real number, then ''f'' is a homomorphism of rings, since ''f'' preserves both addition:
:<math>f(r+s) = \begin{pmatrix}
  r+s & 0 \\
  0 & r+s
\end{pmatrix} = \begin{pmatrix}
  r & 0 \\
  0 & r
\end{pmatrix} + \begin{pmatrix}
  s & 0 \\
  0 & s
\end{pmatrix} = f(r) + f(s)</math>
and multiplication:
:<math>f(rs) = \begin{pmatrix}
  rs & 0 \\
  0 & rs
\end{pmatrix} = \begin{pmatrix}
  r & 0 \\
  0 & r
\end{pmatrix} \begin{pmatrix}
  s & 0 \\
  0 & s
\end{pmatrix} = f(r)\,f(s).</math>
 
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
:''f''(''z'') = |''z''|.
That is, ''&fnof;''(''z'') is the [[absolute value]] (or modulus) of the complex number ''z''.  Then ''f'' is a homomorphism of groups, since it preserves multiplication:
:''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>).
Note that ''&fnof;'' cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
:|''z''<sub>1</sub> + ''z''<sub>2</sub>| ≠ |''z''<sub>1</sub>| + |''z''<sub>2</sub>|.
 
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}}.
 
==Informal discussion==
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''.
 
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.
 
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>}}.
 
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.
 
== Specific kinds of homomorphisms ==<!---renamed, since section is mainly not about category theory--->
[[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.]]
{| style="float:right; border: 1px solid darkgray;"
|-
|
{| class="collapsible collapsed"
|-
! colspan="3" | [Proof 1]
|-
| colspan="3" | For for each ''n''-ary operation ''μ'' and all ''b''<sub>1</sub>,...,''b''<sub>''n''</sub> ∈ ''B'':
|-
| || ''f''<sup>-1</sup>(μ<sub>''B''</sub>(''b''<sub>1</sub>,...,''b''<sub>''n''</sub>))
|-
| = || ''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>))))
| since ''b'' = ''f''(''f''<sup>-1</sup>(''b'')) for each ''b'' ∈ ''B''
|-
| = || ''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>))))
| since ''f'' is a homomorphism
|-
| = || μ<sub>''A''</sub>(''f''<sup>-1</sup>(''b''<sub>1</sub>),...,''f''<sup>-1</sup>(''b''<sub>''n''</sub>))
| since ''a'' = ''f''<sup>-1</sup>(''f''(''a'')) for each ''a'' ∈ ''A''
|}
|-
|
{| class="collapsible collapsed"
|-
! colspan="3" | [Proof 2]
|-
| colspan="3" | If ''g'' is a left inverse of ''f'',
|-
| colspan="3" | and ''f''(''g''<sub>1</sub>(''b'')) = ''f''(''g''<sub>2</sub>(''b'')), then
|-
| || ''g''<sub>1</sub>(''b'')
|-
| = || ''g''(''f''(''g''<sub>1</sub>(''b'')))
| since ''g'' is a left inverse of ''f''
|-
| = || ''g''(''f''(''g''<sub>2</sub>(''b'')))
| since ''f''(''g''<sub>1</sub>(''b'')) = ''f''(''g''<sub>2</sub>(''b''))
|-
| = || ''g''<sub>2</sub>(''b'')
| since ''g'' is a left inverse of ''f''
|}
|-
|
{| class="collapsible collapsed"
|-
! colspan="3" | [Proof 3]
|-
| colspan="3" | If ''g'' is a right inverse of ''f'',
|-
| colspan="3" | and ''g''<sub>1</sub>(''f''(''a'')) = ''g''<sub>2</sub>(''f''(''a'')) for each ''a'' ∈ ''A'', then
|-
| || ''g''<sub>1</sub>(''b'')
|-
| = || ''g''<sub>1</sub>(''f''(''g''(''b'')))
| since ''g'' is a right inverse of ''f''
|-
| = || ''g''<sub>2</sub>(''f''(''g''(''b'')))
| since ''g''(''b'') ∈ ''A''
|-
| = || ''g''<sub>2</sub>(''b'')
| since ''g'' is a right inverse of ''f''
|}
|}
 
<!---sentences moved down: first define kinds, then discuss deviating notions in category theory below--->
<!---for indicating a contrast to category theory notions, the wording "modules and others" is confusing; I suggest "abstract algebra" instead---
In the important special case of [[module homomorphism]]s, and for some other classes of homomorphisms, there are much simpler descriptions, as follows:
--->
In abstract algebra, several specific kinds of homomorphisms are defined as follows:
* An '''[[isomorphism]]''' is a [[bijective]] homomorphism.
* 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''.
* 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''.
* An '''[[endomorphism]]''' is a homomorphism from an algebraic structure to itself.
* An '''[[automorphism]]''' is an endomorphism which is also an isomorphism, i.e., an isomorphism from an algebraic structure to itself.
 
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.
An isomorphism always has an inverse ''f''<sup>−1</sup>, which is a homomorphism, too (cf. Proof 1).
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''.
 
===Relation to category theory===
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.
For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not.
In category theory, a morphism ''f'' : ''A'' → ''B'' is called:
* '''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.)
* '''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.)
* '''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>
<!---deleted, since the notion of "bijective" doesn't appear in the "category theory" article---
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.
--->
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.
 
== Kernel of a homomorphism ==
{{main|Kernel (algebra)}}
 
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]]).
 
== Homomorphisms of relational structures ==
 
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
* ''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'',
* ''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''.
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>
 
==Homomorphisms and e-free homomorphisms in formal language theory==
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''.
 
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.
 
==See also==
* [[continuous function]]
* [[diffeomorphism]]
* [[homomorphic encryption]]
* [[homomorphic secret sharing]] – a simplistic decentralized voting protocol
* [[morphism]]
 
== Notes ==
{{reflist|group=note}}
 
==References==
<div class="references-small">
{{refbegin}}
<references/>
A monograph available free online:
* 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.
</div>
 
[[Category:Morphisms]]

Latest revision as of 06:02, 17 December 2014

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