View source for Homomorphism

{{short description|Structure-preserving map between two algebraic structures of the same type}}
{{distinguish|holomorphism|homeomorphism}}
In [[algebra]], a '''homomorphism''' is a [[morphism|structure-preserving]] [[map (mathematics)|map]] between two [[algebraic structure]]s of the same type (such as two [[group (mathematics)|group]]s, two [[ring (mathematics)|ring]]s, or two [[vector space]]s). The word ''homomorphism'' comes from the [[ancient Greek language]]: ''[[wikt:ὁμός|ὁμός]] (homos)'' meaning "same" and ''[[wikt:μορφή|μορφή]] (morphe)'' meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ''[[wikt:ähnlich|ähnlich]]'' meaning "similar" to ''ὁμός'' meaning "same".<ref>{{cite web|url=https://mathoverflow.net/q/280271 |title=Whence “homomorphism” and “homomorphic”?|author=|date=|website=mathoverflow.net|accessdate=21 March 2018}}</ref>{{OR|date=March 2019}}

Homomorphisms of vector spaces are also called [[linear map]]s, and their study is the object of [[linear algebra]].

The concept of homomorphism has been generalized, under the name of [[morphism]], to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of [[category theory]].

A homomorphism may also be an [[isomorphism]], an [[endomorphism]], an [[automorphism]], etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.

== Definition ==
A homomorphism is a map between two [[algebraic structure]]s of the same type (that is of the same name), that preserves the [[operation (mathematics)|operations]] of the structures. This means a [[map (mathematics)|map]] <math>f: A\to B</math> between two [[set (mathematics)|sets]] {{mvar|A}}, {{mvar|B}} equipped with the same structure such that, if <math>\cdot</math> is an operation of the structure (supposed here, for simplification, to be a [[binary operation]]), then
:<math>f(x\cdot y)=f(x)\cdot f(y)</math>
for every pair {{math|''x'', ''y''}} of elements of {{math|''A''}}.<ref group=note>As it is often the case, but not always, the same symbol for the operation of both {{math|''A''}} and {{math|''B''}} was used here.</ref> One says often that {{math|''f''}} preserves the operation or is compatible with the operation.

Formally, a map <math>f: A\to B</math> preserves an operation {{math|''μ''}} of [[arity]] {{math|''k''}}, defined on both {{math|''A''}} and {{math|''B''}} if 
:<math>f(\mu_A(a_1, \ldots, a_k)) = \mu_B(f(a_1), \ldots, f(a_k)),</math> 
for all elements {{math|''a''<sub>1</sub>, ..., ''a''<sub>''k''</sub>}} in {{math|''A''}}.

The operations that must be preserved by a homomorphism include [[0-ary function|0-ary operations]], that is the constants. In particular, when an [[identity element]] is required by the type of structure, the identity element of the first structure must be mapped to the corresponding identity element of the second structure.

For example:

* A [[semigroup homomorphism]] is a map between [[semigroup]]s that preserves the semigroup operation.
* A [[monoid homomorphism]] is a map between [[monoid]]s that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid (the identity element is a [[0-ary function|0-ary operation]]).
* A [[group homomorphism]] is a map between [[group (mathematics)|groups]] that preserves the group operation. This implies that the group homomorphism maps the identity element of the first group to the identity element of the second group, and maps the [[inverse element|inverse]] of an element of the first group  to the inverse of the image of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism.
* A [[ring homomorphism]] is a map between [[ring (mathematics)|rings]] that preserves the ring addition, the ring multiplication, and the [[multiplicative identity]]. Whether the multiplicative identity is to be preserved depends upon the definition of ''ring'' in use. If the multiplicative identity is not preserved, one has a [[rng (algebra)|rng]] homomorphism.
* A [[linear map]] is a homomorphism of [[vector space]], That is a group homomorphism between vector spaces that preserves the abelian group structure and [[scalar multiplication]].
* A [[module homomorphism]], also called a linear map between [[module (mathematics)|modules]], is defined similarly.
* An [[algebra homomorphism]] is a map that preserves the [[algebra over a field|algebra]] operations.

An algebraic structure may have more than one operation, and a homomorphism is required to preserve each operation. Thus a map that preserves only some of the operations is not a homomorphism of the structure, but only a homomorphism of the substructure obtained by considering only the preserved operations. For example, a map between monoids that preserves the monoid operation and not the identity element, is not a monoid homomorphism, but only a semigroup homomorphism.

The notation for the operations does not need to be the same in the source and the target of a homomorphism. For example, the [[real number]]s form a group for addition, and the positive real numbers form a group for multiplication. The [[exponential function]] 
:<math>x\mapsto e^x</math>
satisfies
:<math>e^{x+y} = e^xe^y,</math>
and is thus a homomorphism between these two groups. It is even an isomorphism (see below), as its [[inverse function]], the [[natural logarithm]], satisfies 
:<math>\ln(xy)=\ln(x)+\ln(y), </math>
and is also a group homomorphism.

== Examples ==

[[File:Exponentiation as monoid homomorphism svg.svg|thumb|x200px|[[Monoid]] homomorphism {{math|''f''}} from the monoid {{math|{{color|#008000|('''N''', +, 0)}}}} to the monoid {{math|{{color|#800000|('''N''', ×, 1)}}}}, defined by {{math|1=''f''(''x'') = 2<sup>''x''</sup>}}. It is [[Injective function|injective]], but not [[Surjective function|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 {{mvar|r}} is a real number, then {{mvar|f}} is a homomorphism of rings, since {{mvar|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 {{math|''f''}} from the nonzero complex numbers to the nonzero real numbers by
:<math>f(z) = |z| .</math>
That is, {{math|''f''(''z'')}} is the [[absolute value]] (or modulus) of the complex number {{math|''z''}}.  Then {{math|''f''}} is a homomorphism of groups, since it preserves multiplication:
:<math>f(z_1 z_2) = |z_1 z_2| = |z_1| |z_2| = f(z_1) f(z_2) .</math>
Note that {{math|''f''}} cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
:<math>|z_1 + z_2| \ne |z_1| + |z_2| .</math>

As another example, the diagram shows a [[monoid]] homomorphism {{math|''f''}} from the monoid {{math|('''N''', +, 0)}} to the monoid {{math|('''N''', ×, 1)}}. Due to the different names of corresponding operations, the structure preservation properties satisfied by {{math|''f''}} amount to {{math|1=''f''(''x'' + ''y'') = ''f''(''x'') × ''f''(''y'')}} and {{math|1=''f''(0) = 1}}.

A [[composition algebra]] {{math|''A''}} over a field {{math|''F''}} has a [[quadratic form]], called a ''norm'', {{math|''N'' : ''A'' → ''F''}}, which is a group homomorphism from the [[multiplicative group]] of {{math|''A''}} to the multiplicative group of {{math|''F''}}.

== Special homomorphisms ==
Several kinds of homomorphisms have a specific name, which is also defined for general [[morphism]]s.

=== Isomorphism ===
An [[isomorphism]] between [[algebraic structure]]s of the same type is commonly defined as a [[bijective]] homomorphism.<ref name="Birkhoff.1967">{{Citation | last1=Birkhoff | first1=Garrett | title=Lattice theory | origyear=1940 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=3rd | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-1025-5 | mr=598630 | year=1967 | volume=25}}</ref>{{rp|134}} <ref name="Burris.Sankappanavar.2012">{{cite book | url=http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf | isbn=978-0-9880552-0-9 | author1=Stanley N. Burris | author2=H.P. Sankappanavar | title=A Course in Universal Algebra | year=2012 }}</ref>{{rp|28}}

In the more general context of [[category theory]], an isomorphism is defined as a [[morphism]], which has an [[inverse function|inverse]] that is also a morphism. In the specific case of algebraic structures, the two definitions are equivalent, although they may differ for non-algebraic structures, which have an underlying set.

More precisely, if 
:<math>f: A\to B</math>
is a (homo)morphism, it has an inverse if there exists a homomorphism
:<math>g: B\to A</math>
such that
:<math>f\circ g = \operatorname{Id}_B \qquad \text{and} \qquad g\circ f = \operatorname{Id}_A.</math>

If {{math|''A''}} and {{math|''B''}} have underlying sets, and <math>f: A\to B</math> has an inverse {{math|''g''}}, then {{math|''f''}} is bijective. In fact, {{math|''f''}} is [[injective]], as {{math|1=''f''(''x'') = ''f''(''y'')}} implies {{math|1=''x'' = ''g''(''f''(x)) = ''g''(''f''(y)) = ''y''}}, and  {{math|''f''}} is [[surjective]], as, for any {{math|''x''}} in {{math|''B''}}, one has  {{math|1=''x'' = ''f''(''g''(''x''))}}, and {{math|''x''}} is the image of an element of {{math|''A''}}.

Conversely, if <math>f: A\to B</math> is a bijective homomorphism between algebraic structures, let <math>g: B\to A</math> be the map such that {{math|''g''(''y'')}} is the unique element {{math|''x''}} of {{math|''A''}} such that {{math|1=''f''(''x'') = ''y''}}. One has <math>f\circ g = \operatorname{Id}_B \text{ and } g\circ f = \operatorname{Id}_A,</math> and it remains only to show that {{math|''g''}} is a homomorphism. If <math>*</math> is a binary operation of the structure, for every pair {{math|''x'', ''y''}} of elements of {{math|''B''}}, one has
:<math>g(x*_B y) = g(f(g(x))*_Bf(g(y))) = g(f(g(x)*_A g(y))) = g(x)*_A g(y),</math>
and {{math|''g''}} is thus compatible with <math>*.</math> As the proof is similar for any [[arity]], this shows that {{math|''g''}} is a homomorphism.

This proof does not work for non-algebraic structures. For examples, for [[topological space]]s, a morphism is a [[continuous map]], and the inverse of a bijective continuous map is not necessarily continuous. An isomorphism of topological spaces, called [[homeomorphism]] or [[bicontinuous function|bicontinuous map]], is thus a bijective continuous map, whose inverse is also continuous.

===Endomorphism===
An [[endomorphism]] is a homomorphism whose [[domain of a function|domain]] equals the [[codomain]], or, more generally, a [[morphism]] whose source is equal to the target.<ref name="Birkhoff.1967"/>{{rp|135}}

The endomorphisms of an algebraic structure, or of an object of a [[category (mathematics)|category]] form a [[monoid]] under composition.

The endomorphisms of a [[vector space]] or of a [[module (mathematics)|module]] form a [[ring (mathematics)|ring]]. In the case of a vector space or a [[free module]] of finite [[dimension (vector space)|dimension]], the choice of a [[basis (vector space)|basis]] induces a [[ring isomorphism]] between the ring of endomorphisms and the ring of [[square matrices]] of the same dimension.

===Automorphism===

An [[automorphism]] is an endomorphism that is also an isomorphism.<ref name="Birkhoff.1967"/>{{rp|135}}

The automorphisms of an algebraic structure or of an object of a category form a [[group (mathematics)|group]] under composition, which is called the [[automorphism group]] of the structure.

Many groups that have received a name are automorphism groups of some algebraic structure. For example, the [[general linear group]] <math>\operatorname{GL}_n(k)</math> is the automorphism group of a [[vector space]] of dimension {{math|''n''}} over a [[field (mathematics)|field]] {{math|''k''}}.

The automorphism groups of [[field (mathematics)|field]]s were introduced by [[Évariste Galois]] for studying the [[root of a polynomial|roots]] of [[polynomial]]s, and are the basis of [[Galois theory]].

===Monomorphism===
For algebraic structures, [[monomorphism]]s are commonly defined as [[injective]] homomorphisms.<ref name="Birkhoff.1967"/>{{rp|134}} <ref name="Burris.Sankappanavar.2012"/>{{rp|29}}

In the more general context of [[category theory]], a monomorphism is defined as a [[morphism]] that is '''[[Cancellation property|left cancelable]]'''.<ref name=workmath>{{cite book | at=Exercise 4 in section I.5 | first=Saunders | last=Mac Lane| authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | volume=5 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | isbn=0-387-90036-5 | year=1971 | zbl=0232.18001 }}</ref> This means that a (homo)morphism <math>f:A\to B</math> is a monomorphism if, for any pair {{math|''g'', ''h''}} of morphisms from any other object {{math|''C''}} to {{math|''A''}}, then <math>f\circ g = f\circ h</math> implies {{math|1=''g'' = ''h''}}.

These two definitions of ''monomorphism'' are equivalent for all common algebraic structures. More precisely, they are equivalent for [[field (mathematics)|fields]], for which every homomorphism is a monomorphism, and for [[variety (universal algebra)|varieties]] of [[universal algebra]], that is algebraic structures for which operations and axioms (identities) are defined without any restriction (fields are not a variety, as the [[multiplicative inverse]] is defined either as a [[unary operation]] or as a property of the multiplication, which are, in both cases, defined only for nonzero elements).

In particular, the two definitions of a monomorphism are equivalent for [[set (mathematics)|sets]], [[magma (algebra)|magmas]], [[semigroup]]s, [[monoid]]s, [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]], [[vector space]]s and [[module (mathematics)|modules]].

A '''[[split monomorphism]]''' is a homomorphism that has a [[inverse function#Left and right inverses|left inverse]] and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism <math>f\colon A \to B</math> is a split homomorphism if there exists a homomorphism <math>g\colon B \to A</math> such that <math>g\circ f = \operatorname{Id}_A.</math> A split monomorphism is always a monomorphism, for both meanings of ''monomorphism''. For sets and vector spaces, every monomorphism is a split homomorphism, but this property does not hold for most common algebraic structures.

{{cot|Proof of the equivalence of the two definitions of monomorphisms}}
''An injective homomorphism is left cancelable'': If <math>f\circ g = f\circ h,</math> one has <math>f(g(x))=f(h(x))</math> for every {{math|''x''}} in {{math|''C''}}, the common source of {{math|''g''}} and {{math|''h''}}. If {{math|''f''}} is injective, then {{math|1=''g''(''x'') = ''h''(''x'')}}, and thus {{math|1=''g'' = ''h''}}. This proof works not only for algebraic structures, but also for any [[category (mathematics)|category]] whose objects are sets and arrows are maps between these sets. For example, an injective continuous map is a monomorphism in the category of [[topological space]]s.

For proving that, conversely, a left cancelable homomorphism is injective, it is useful to consider a ''[[free object]] on {{math|x}}''. Given a [[variety (universal algebra)|variety]] of algebraic structures a free object on {{math|''x''}} is a pair consisting of an algebraic structure {{math|''L''}} of this variety and an element {{math|''x''}} of {{math|''L''}} satisfying the following [[universal property]]: for every structure {{math|''S''}} of the variety, and every element {{math|''s''}} of {{math|''S''}}, there is a unique homomorphism <math>f: L\to S</math> such that {{math|1=''f''(''x'') = ''s''}}. For example, for sets, the free object on {{math|''x''}} is simply {{math|{''x''}{{void}}}}; for [[semigroup]]s, the free object on {{math|''x''}} is <math>\{x, x^2, \ldots, x^n, \ldots\},</math> which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for [[monoid]]s, the free object on {{math|''x''}} is <math>\{1, x, x^2, \ldots, x^n, \ldots\},</math> which, as, a monoid, is isomorphic to the additive monoid of the nonnegative integers; for [[group (mathematics)|group]]s, the free object on {{math|''x''}} is the [[infinite cyclic group]] <math>\{\ldots, x^{-n}, \ldots, x^{-1}, 1, x, x^2, \ldots, x^n, \ldots\},</math> which, as, a group, is isomorphic to the additive group of the integers; for [[ring (mathematics)|rings]], the free object on {{math|''x''}} is the [[polynomial ring]] <math>\mathbb{Z}[x];</math> for [[vector space]]s or [[module (mathematics)|modules]], the free object on {{math|''x''}} is the vector space or free module that has {{math|''x''}} as a basis.

''If a free object over {{math|x}} exists, then every left cancelable homomorphism is injective'': let <math>f\colon A \to B</math> be a left cancelable homomorphism, and {{math|''a''}} and {{math|''b''}} be two elements of {{math|''A''}} such {{math|1=''f''(''a'') = ''f''(''b'')}}. By definition of the free object {{math|''F''}}, there exist homomorphisms {{math|''g''}} and {{math|''h''}} from {{math|''F''}} to {{math|''A''}} such that {{math|1=''g''(''x'') = ''a''}} and {{math|1=''h''(''x'') = ''b''}}. As {{math|1=''f''(''g''(''x'') = ''f''(''h''(''x'') }}, one has <math>f\circ g = f\circ h, </math> by the uniqueness in the definition of a universal property. As {{math|''f''}} is left cancelable, one has {{math|1=''g'' = ''h''}}, and thus {{math|1=''a'' = ''b''}}. Therefore, {{math|''f''}} is injective.

''Existence of a free object on {{math|x}} for a [[variety (universal algebra)|variety]]'' (see also {{slink|Free object|Existence}}): For building a free object over {{math|''x''}}, consider the set {{math|''W''}} of the [[well-formed formula]]s built up from {{math|''x''}} and the operations of the structure. Two such formulas are said equivalent if one may pass from one to the other by applying the axioms ([[identity (mathematics)|identities]] of the structure). This defines an [[equivalence relation]], if the identities are not subject to conditions, that is if one works with a variety. Then the operations of the variety are well defined on the set of [[equivalence class]]es of {{math|''W''}} for this relation. It is straightforward to show that the resulting object is a free object on {{math|''W''}}.
{{cob}}

===Epimorphism===

In [[algebra]], '''epimorphisms''' are often defined as [[surjective]] homomorphisms.<ref name="Birkhoff.1967"/>{{rp|134}}<ref name="Burris.Sankappanavar.2012" />{{rp|43}} On the other hand, in [[category theory]], [[epimorphism]]s are defined as '''right cancelable''' [[morphism]]s.<ref name=workmath/> This means that a (homo)morphism <math>f:A\to B</math> is an epimorphism if, for any pair {{math|''g'', ''h''}} of morphisms from {{math|''B''}} to any other object {{math|''C''}}, the equality <math>g\circ f = h\circ f</math> implies {{math|1=''g'' = ''h''}}.

A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures. However, the two definitions of ''epimorphism'' are equivalent for [[set (mathematics)|sets]], [[vector space]]s, [[abelian group]]s, [[module (mathematics)|modules]] (see below for a proof), and [[group (mathematics)|groups]].<ref>Linderholm, C. E. (1970). A group epimorphism is surjective. ''The American Mathematical Monthly'', 77(2), 176-177.</ref> The importance of these structures in all mathematics, and specially in [[linear algebra]] and [[homological algebra]], may explain the coexistence of two non-equivalent definitions.

Algebraic structures for which there exist non-surjective epimorphisms include [[semigroup]]s and [[ring (mathematics)|rings]]. The most basic example is the inclusion of [[integer]]s into [[rational number]]s, which is an homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism.<ref name=workmath/><ref>{{cite book | page=363 | title=Hopf Algebra: An Introduction | zbl=0962.16026 | series=Pure and Applied Mathematics | volume=235 | location=New York, NY | publisher=Marcel Dekker | first1=Sorin | last1=Dăscălescu | first2=Constantin | last2=Năstăsescu | first3=Şerban | last3=Raianu | year=2001 | isbn=0824704819 }}</ref>

A wide generalization of this example is the [[localization of a ring]] by a multiplicative set. Every localization is a ring epimorphism, which is not, in general, surjective. As localizations are fundamental in [[commutative algebra]] and [[algebraic geometry]], this may explain why in these areas, the definition of epimorphisms as right cancelable homomorphisms is generally preferred.

A '''[[split epimorphism]]''' is a homomorphism that has a [[inverse function#Left and right inverses|right inverse]] and thus it is itself a left inverse of that other homomorphism. That is, a homomorphism <math>f\colon A \to B</math> is a split epimorphism if there exists a homomorphism <math>g\colon B \to A</math> such that <math>f\circ g = \operatorname{Id}_B.</math> A split epimorphism is always an epimorphism, for both meanings of ''epimorphism''. For sets and vector spaces, every epimorphism is a split epimorphism, but this property does not hold for most common algebraic structures.

In summary, one has 
:<math>\text {split epimorphism} \implies \text{epimorphism (surjective)}\implies \text {epimorphism (right cancelable)};</math> 
the last implication is an equivalence for sets, vector spaces, modules and abelian groups; the first implication is an equivalence for sets and vector spaces.
{{cot|Equivalence of the two definitions of epimorphism}}
Let <math>f\colon A \to B</math> be a homomorphism. We want to prove that if it is not surjective, it is not right cancelable.

In the case of sets, let {{math|''b''}} be an element of {{math|''B''}} that not belongs to {{math|''f''(''A'')}}, and define <math>g,h\colon B\to B</math> such that {{math|''g''}} is the [[identity function]], and that {{math|1=''h''(''x'') = ''x''}} for every <math>x\in B,</math> except that {{math|''h''(''b'')}} is any other element of {{math|''B''}}. Clearly {{math|''f''}} is not right cancelable, as {{math|''g'' ≠ ''h''}} and <math>g\circ f = h\circ f.</math>

In the case of vector spaces, abelian groups and modules, the proof relies on the existence of [[cokernel]]s and on the fact that the [[zero map]]s are homomorphisms: let {{math|''C''}} be the cokernel of {{math|''f''}}, and <math>g\colon B\to C</math> be the canonical map, such that {{math|1=''g''(''f''(''A'')) = 0}}. Let <math>h\colon B\to C</math> be the zero map. If {{math|''f''}} is not surjective, {{math|''C'' ≠ 0}}, and thus {{math|''g'' ≠ ''h''}} (one is a zero map, while the other is not). Thus {{math|''f''}} is not cancelable, as <math>g\circ f = h\circ f</math> (both are the zero map from {{math|''A''}} to {{math|''C''}}).
{{cob}}

== Kernel ==
{{main|Kernel (algebra)}}

Any homomorphism {{math|''f'' : ''X'' → ''Y''}} defines an [[equivalence relation]] {{math|~}} on {{math|''X''}} by {{math|''a'' ~ ''b''}} if and only if {{math|1=''f''(''a'') = ''f''(''b'')}}. The relation {{math|~}} is called the '''kernel''' of {{math|''f''}}. It is a [[congruence relation]] on {{math|''X''}}. The [[quotient set]] {{math|''X'' / ~}} can then be given a structure of the same type as {{mvar|X}}, in a natural way, by defining the operations of the quotient set by {{math|1=[''x''] ∗ [''y''] =  [''x'' ∗ ''y'']}}, for each operation {{math|∗}} of {{math|''X''}}. In that case the image of {{math|''X''}} in {{math|''Y''}} under the homomorphism {{math|''f''}} is necessarily [[isomorphic]] to {{math|''X'' / ~}}; this fact is one of the [[isomorphism theorem]]s.

When the algebraic structure is a  [[group (mathematics)|group]] for some operation, the [[equivalence class]] {{math|''K''}} of the [[identity element]] of this operation suffices to characterize the equivalence relation. In this case, the quotient by the equivalence relation is denoted by {{math|''X''/''K''}} (usually read as "{{math|''X''}} [[Ideal (ring theory)|mod]] {{math|''K''}}"). Also in this case, it is {{math|''K''}}, rather than {{math|~}}, that is called the [[kernel (algebra)|kernel]] of {{math|''f''}}. The kernels of homomorphisms of a given type of algebraic structure are naturally equipped with some structure. This structure type of the kernels is the same as the considered structure, in the case of [[abelian group]]s, [[vector space]]s and [[module (mathematics)|modules]], but is different and has received a specific name in other cases, such as [[normal subgroup]] for kernels of [[group homomorphisms]] and [[ideal (ring theory)|ideals]] for kernels of [[ring homomorphism]]s (in the case of non-commutative rings, the kernels are the [[two-sided ideal]]s).

== 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>

==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> and are often 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'' on Σ<sub>1</sub><sup>∗</sup>.<ref group=note>The ∗ denotes the [[Kleene star]] operation, while Σ<sup>∗</sup> denotes the set of words formed from the alphabet Σ, including the empty word. Juxtaposition of terms denotes [[concatenation]]. For example, ''h''(''u'') ''h''(''v'') denotes the concatenation of ''h''(''u'') with ''h''(''v'').</ref> If ''h'' is a homomorphism on Σ<sub>1</sub><sup>∗</sup> and ''e'' denotes the empty word, then ''h'' is called an ''e-free homomorphism'' when {{nowrap|''h''(''x'') ≠ ''e''}} for all {{nowrap|''x'' ≠ ''e''}} in Σ<sub>1</sub><sup>∗</sup>.

The set Σ<sup>∗</sup> of words formed from the alphabet Σ may be thought of as the [[free monoid]] generated by Σ. Here the monoid operation is concatenation and the identity element is the empty word. From this perspective, a language homormorphism is precisely a monoid homomorphism.<ref group=note>We are assured that a language homomorphism ''h'' maps the empty word ''e'' to the empty word. Since ''h''(''e'') = ''h''(''ee'') = ''h''(''e'')''h''(''e''), the number ''w'' of characters in ''h''(''e'') equals the number 2''w'' of characters in ''h''(''e'')''h''(''e''). Hence ''w'' = 0 and ''h''(''e'') has null length.</ref>

==See also==
* [[Continuous function]]
* [[Diffeomorphism]]
* [[Homomorphic encryption]]
* [[Homomorphic secret sharing]] – a simplistic decentralized voting protocol
* [[Morphism]]

== Notes ==
{{reflist|group=note}}

==Citations==
<references/>

==References==
*{{cite book | url=http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf | isbn=978-0-9880552-0-9 | author1=Stanley N. Burris | author2=H.P. Sankappanavar | title=A Course in Universal Algebra | year=2012 }}
*{{citation | first1 = Saunders | last1 = Mac Lane| authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | volume=5 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | isbn=0-387-90036-5 | year=1971 | zbl=0232.18001 }}
*{{citation | first1 = John B. | last1 = Fraleigh | first2 = Victor J. | last2 = Katz | year = 2003 | title = A First Course in Abstract Algebra | publisher = Addison-Wesley | isbn= 978-1-292-02496-7}}

[[Category:Morphisms]]