Editing 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-lang|grc|ὁμός}} ({{transliteration|grc|homos}}) meaning "same" and {{wikt-lang|grc|μορφή}} ({{transliteration|grc|morphe}}) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German {{wikt-lang|de|ähnlich}} meaning "similar" to {{lang|grc|ὁμός}} meaning "same".<ref>{{Cite book|last=Fricke|first=Robert|url=https://archive.org/details/vorlesungenber01fricuoft/page/n5/mode/2up|title=Vorlesungen über die Theorie der automorphen Functionen|language=de|date=1897–1912|publisher=B. G. Teubner|oclc=29857037}}</ref> The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician [[Felix Klein]] (1849–1925).<ref>See:
*  {{cite journal |last1=Ritter |first1=Ernst |title=Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze |language=de |journal=Mathematische Annalen |date=1892 |volume=41 |pages=1–82 |doi=10.1007/BF01443449 |s2cid=121524108 |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102918109&view=1up&seq=15 |trans-title=The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem |quote=[footnote p. 22:] Ich will nach einem Vorschlage von Hrn. Prof. Klein statt der umständlichen und nicht immer ausreichenden Bezeichnungen: 'holoedrisch, bezw. hemiedrisch u.s.w. isomorph' die Benennung 'isomorph' auf den Fall des ''holoedrischen'' Isomorphismus zweier Gruppen einschränken, sonst aber von 'Homomorphismus' sprechen, ...|trans-quote=Following a suggestion of Prof. Klein, instead of the cumbersome and not always satisfactory designations "holohedric, or hemihedric, etc. isomorphic", I will limit the denomination "isomorphic" to the case of a ''holohedric'' isomorphism of two groups; otherwise, however, [I will] speak of a "homomorphism", ...}}
*  {{cite journal |last1=Fricke |first1=Robert |title=Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen |language=de |journal=Mathematische Annalen |date=1892 |volume=41 |issue=3 |pages=443–468 |doi=10.1007/BF01443421 |s2cid=120022176 |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102918109&view=1up&seq=471 |trans-title=On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7) |quote=[p. 466] Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe Γ<sub>(63)</sub> auf die Gruppe der mod. n incongruenten Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet. ... *) Im Anschluss an einen von Hrn. Klein bei seinen neueren Vorlesungen eingeführten Brauch schreibe ich an Stelle der bisherigen Bezeichnung 'meroedrischer Isomorphismus' die sinngemässere 'Homomorphismus'.|trans-quote=Thus, as one immediately sees, a homomorphic relation of the group Γ<sub>(63)</sub> is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1. ... Following a usage that has been introduced by Mr. Klein during his more recent lectures, I write in place of the earlier designation 'merohedral isomorphism' the more logical 'homomorphism'.}}</ref>

Homomorphisms of vector spaces are also called [[linear map]]s, and their study is the subject 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 (e.g. two groups, two fields, two vector spaces), 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]] <math>A</math>, <math>B</math> 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 display="block">f(x\cdot y)=f(x)\cdot f(y)</math>

for every pair <math>x</math>, <math>y</math> of elements of <math>A</math>.<ref group="note">As it is often the case, but not always, the same symbol for the operation of both <math>A</math> and <math>B</math> was used here.</ref> One says often that <math>f</math> preserves the operation or is compatible with the operation.

Formally, a map <math>f: A\to B</math> preserves an operation <math>\mu</math> of [[arity]] <math>k</math>, defined on both <math>A</math> and <math>B</math> if 

<math display="block">f(\mu_A(a_1, \ldots, a_k)) = \mu_B(f(a_1), \ldots, f(a_k)),</math> 

for all elements <math>a_1, ..., a_k</math> in <math>A</math>.

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]]s; 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 display="block">x\mapsto e^x</math>

satisfies

<math display="block">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 display="block">\ln(xy)=\ln(x)+\ln(y), </math>
and is also a group homomorphism.

== Examples ==

[[File:Exponentiation as monoid homomorphism svg.svg|thumb|[[Monoid]] homomorphism <math>f</math> from the monoid {{math|{{color|#008000|('''N''', +, 0)}}}} to the monoid {{math|{{color|#800000|('''N''', ×, 1)}}}}, defined by <math>f(x) = 2^x</math>. 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 2×2 [[matrix (mathematics)|matrices]] is also a ring, under [[matrix addition]] and [[matrix multiplication]].  If we define a function between these rings as follows:

<math display="block">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 display="block">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 display="block">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</math> from the nonzero complex numbers to the nonzero real numbers by

<math display="block">f(z) = |z| .</math>

That is, <math>f</math> is the [[absolute value]] (or modulus) of the complex number <math>z</math>.  Then <math>f</math> is a homomorphism of groups, since it preserves multiplication:

<math display="block">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 display="block">|z_1 + z_2| \ne |z_1| + |z_2|.</math>

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

A [[composition algebra]] <math>A</math> over a field <math>F</math> has a [[quadratic form]], called a ''norm'', <math>N: A \to F</math>, which is a group homomorphism from the [[multiplicative group]] of <math>A</math> to the multiplicative group of <math>F</math>.

== Special homomorphisms ==
Several kinds of homomorphisms have a specific name, which is also defined for general [[morphism]]s.
[[File:Venn Diagram of Homomorphisms.jpg|thumb|General relationship of homomorphisms (including [[inner automorphism]]s, labelled as "Inner").]]

=== Isomorphism ===
An [[isomorphism]] between [[algebraic structure]]s of the same type is commonly defined as a [[bijective]] homomorphism.<ref name="Birkhoff.1967">{{cite book | last1=Birkhoff | first1=Garrett | title=Lattice theory | orig-year=1940 | publisher=[[American Mathematical Society]] | location=Providence, Rhode Island | 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 | first1=Stanley N. |last1=Burris | first2=H. P. |last2=Sankappanavar | title=A Course in Universal Algebra | year=2012 | publisher=S. Burris and H.P. Sankappanavar }}</ref>{{rp|28}}

In the more general context of [[category theory]], an isomorphism is defined as a [[morphism]] that 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 display="block">f: A\to B</math>

is a (homo)morphism, it has an inverse if there exists a homomorphism

<math display="block">g: B\to A</math>

such that

<math display="block">f\circ g = \operatorname{Id}_B \qquad \text{and} \qquad g\circ f = \operatorname{Id}_A.</math>

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

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)</math> is the unique element <math>x</math> of <math>A</math> such that <math>f(x) = y</math>. 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</math>, <math>y</math> of elements of <math>B</math>, one has

<math display="block">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</math> is thus compatible with <math>*.</math> As the proof is similar for any [[arity]], this shows that <math>g</math> is a homomorphism.

This proof does not work for non-algebraic structures. For example, 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 its 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</math> over a [[field (mathematics)|field]] <math>k</math>.

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| author-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | volume=5 | series=[[Graduate Texts in Mathematics]] | publisher=Springer | 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</math>, <math>h</math> of morphisms from any other object <math>C</math> to <math>A</math>, then <math>f \circ g = f \circ h</math> implies <math>g = h</math>.

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 (the fields do not form 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 monomorphism 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 monomorphism, but this property does not hold for most common algebraic structures.

{{collapse top|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</math> in <math>C</math>, the common source of <math>g</math> and <math>h</math>. If <math>f</math> is injective, then <math>g(x) = h(x)</math>, and thus <math>g = h</math>. 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</math>''. Given a [[variety (universal algebra)|variety]] of algebraic structures a free object on <math>x</math> is a pair consisting of an algebraic structure <math>L</math> of this variety and an element <math>x</math> of <math>L</math> satisfying the following [[universal property]]: for every structure <math>S</math> of the variety, and every element <math>s</math> of <math>S</math>, there is a unique homomorphism <math>f: L\to S</math> such that <math>f(x) = s</math>. For example, for sets, the free object on <math>x</math> is simply <math>\{x\}</math>; for [[semigroup]]s, the free object on <math>x</math> 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</math> 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</math> 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</math> is the [[polynomial ring]] <math>\mathbb{Z}[x];</math> for [[vector space]]s or [[module (mathematics)|modules]], the free object on <math>x</math> is the vector space or free module that has <math>x</math> as a basis.

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

''Existence of a free object on <math>x</math> for a [[variety (universal algebra)|variety]]'' (see also {{slink|Free object|Existence}}): For building a free object over <math>x</math>, consider the set <math>W</math> of the [[well-formed formula]]s built up from <math>x</math> 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</math> for this relation. It is straightforward to show that the resulting object is a free object on <math>x</math>.
{{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</math>, <math>h</math> of morphisms from <math>B</math> to any other object <math>C</math>, the equality <math>g \circ f = h \circ f</math> implies <math>g = h</math>.

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>{{cite journal |last=Linderholm |first=C. E. |year=1970 |title=A group epimorphism is surjective |journal=The American Mathematical Monthly |volume=77 |issue=2 |pages=176–177|doi=10.1080/00029890.1970.11992448 }}</ref> The importance of these structures in all mathematics, especially 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 a 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 City | publisher=Marcel Dekker | first1=Sorin | last1=Dăscălescu | first2=Constantin | last2=Năstăsescu | first3=Șerban | 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 display="block">\text {split epimorphism} \implies \text{epimorphism (surjective)}\implies \text {epimorphism (right cancelable)};</math>

the last implication is an equivalence for sets, vector spaces, modules, abelian groups, and groups; the first implication is an equivalence for sets and vector spaces.
{{collapse top|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</math> be an element of <math>B</math> that not belongs to <math>f(A)</math>, and define <math>g, h\colon B \to B</math> such that <math>g</math> is the [[identity function]], and that <math>h(x) = x</math> for every <math>x \in B,</math> except that <math>h(b)</math> is any other element of <math>B</math>. Clearly <math>f</math> is not right cancelable, as <math>g \neq h</math> 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</math> be the cokernel of <math>f</math>, and <math>g\colon B \to C</math> be the canonical map, such that <math>g(f(A)) = 0</math>. Let <math>h\colon B\to C</math> be the zero map. If <math>f</math> is not surjective, <math>C \neq 0</math>, and thus <math>g \neq h</math> (one is a zero map, while the other is not). Thus <math>f</math> is not cancelable, as <math>g \circ f = h \circ f</math> (both are the zero map from <math>A</math> to <math>C</math>).
{{cob}}

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

Any homomorphism <math>f: X \to Y</math> defines an [[equivalence relation]] <math>\sim</math> on <math>X</math> by <math>a \sim b</math> if and only if <math>f(a) = f(b)</math>. The relation <math>\sim</math> is called the '''kernel''' of <math>f</math>. It is a [[congruence relation]] on <math>X</math>. The [[quotient set]] <math>X/{\sim}</math> can then be given a structure of the same type as <math>X</math>, in a natural way, by defining the operations of the quotient set by <math>[x] \ast [y] =  [x \ast y]</math>, for each operation <math>\ast</math> of <math>X</math>. In that case the image of <math>X</math> in <math>Y</math> under the homomorphism <math>f</math> is necessarily [[isomorphic]] to <math>X/\!\sim</math>; 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</math> 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</math> (usually read as "<math>X</math> [[Ideal (ring theory)|mod]] <math>K</math>"). Also in this case, it is <math>K</math>, rather than <math>\sim</math>, that is called the [[kernel (algebra)|kernel]] of <math>f</math>. 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]].<ref>For a detailed discussion of relational homomorphisms and isomorphisms see {{cite book |at=Section 17.3 |first=Gunther |last=Schmidt |author-link=Gunther Schmidt |year=2010 |title=Relational Mathematics |publisher=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>{{cite book |first=Seymour |last=Ginsburg |author-link=Seymour Ginsburg |title=Algebraic and automata theoretic properties of formal languages |publisher=North-Holland |year=1975 |isbn=0-7204-2506-9}}</ref> and are often briefly referred to as ''morphisms''.<ref>{{cite book |first1=T. |last1=Harju |first2=J. |last2=Karhumӓki |chapter=Morphisms |title=Handbook of Formal Languages |volume=I |editor1-first=G. |editor1-last=Rozenberg |editor2-first=A. |editor2-last=Salomaa |publisher=Springer |year=1997 |isbn=3-540-61486-9}}</ref> Given alphabets <math>\Sigma_1</math> and <math>\Sigma_2</math>, a function <math>h \colon \Sigma_1^* \to \Sigma_2^*</math> such that <math>h(uv) = h(u) h(v)</math> for all <math>u,v \in \Sigma_1</math> is called a ''homomorphism'' on <math>\Sigma_1^*</math>.<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 <math>h</math> is a homomorphism on <math>\Sigma_1^*</math> and <math>\varepsilon</math> denotes the empty string, then <math>h</math> is called an <math>\varepsilon</math>''-free homomorphism'' when <math>h(x) \neq \varepsilon</math> for all <math>x \neq \varepsilon</math> in <math>\Sigma_1^*</math>.

A homomorphism <math>h \colon \Sigma_1^* \to \Sigma_2^*</math> on <math>\Sigma_1^*</math> that satisfies <math>|h(a)| = k</math> for all <math>a \in \Sigma_1</math> is called a <math>k</math>''-uniform'' homomorphism.{{sfn|Krieger|2006|p=287}} If <math>|h(a)| = 1</math> for all <math>a \in \Sigma_1</math> (that is, <math>h</math> is 1-uniform), then <math>h</math> is also called a ''coding'' or a ''projection''.{{citation needed|reason=Give an example citation for each synonym.|date=July 2022}}

The set <math>\Sigma^*</math> of words formed from the alphabet <math>\Sigma</math> may be thought of as the [[free monoid]] generated by {{nowrap|<math>\Sigma</math>.}} Here the monoid operation is [[concatenation]] and the identity element is the empty word. From this perspective, a language homomorphism is precisely a monoid homomorphism.<ref group=note>We are assured that a language homomorphism ''h'' maps the empty word ''ε'' to the empty word. Since ''h''(''ε'') = ''h''(''εε'') = ''h''(''ε'')''h''(''ε''), the number ''w'' of characters in ''h''(''ε'') equals the number 2''w'' of characters in ''h''(''ε'')''h''(''ε''). Hence ''w'' = 0 and ''h''(''ε'') has null length.</ref>

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

== Notes ==
{{Reflist|group=note}}

==Citations==
{{Reflist}}

==References==
* {{cite book |last=Krieger |first=Dalia |editor-last1=Ibarra |editor-first1=Oscar H. |editor-last2=Dang |editor-first2=Zhe | contribution=On critical exponents in fixed points of non-erasing morphisms | title=Developments in language theory : 10th international conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006 : proceedings | publisher=Springer | publication-place=Berlin | pages=280–291 | date=2006 | isbn=978-3-540-35430-7 | oclc=262693179}}
*{{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 | publisher=S. Burris and H.P. Sankappanavar }}
*{{citation | first1 = Saunders | last1 = Mac Lane| author-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | volume=5 | series=[[Graduate Texts in Mathematics]] | publisher=Springer | 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}}

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[[Category:Morphisms]]