Editing Homomorphism (section)

== 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>.
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===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>).
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