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In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape".

Homomorphisms of vector spaces are also called linear maps, 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 a underlying set, or are not algebraic. This generalization is the starting point of category theory.

Isomorphisms, automorphisms, and endomorphisms are special types of morphisms, and thus of homomorphisms.

Definition

A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures. This means a map $f:A\to B$ between two sets $A$ , $B$ equipped with the same structure such that, if $*$ is an operation of the structure (supposed here, for simplification, to be a binary operation), then

f(x*y)=f(x)*f(y)

for every pair $x, y$ of elements of $A$ .^{[note 1]} One says often that $f$ preserves the operation or is compatible with the operation.

Formally, a map $f:A\to B$ preserves an operation $μ$ of arity $k$ , defined on both $A$ and $B$ if

f(\mu _{A}(a_{1},\ldots ,a_{k}))=\mu _{B}(f(a_{1}),\ldots ,f(a_{k})),

for all elements $a 1, ..., a k$ in $A$ .

For example:

A semigroup homomorphism is a map between semigroups that preserves the semigroup operation.
A monoid homomorphism is a map between monoids 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 operation).
A group homomorphism is a map between 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 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 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 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 modules, is defined similarly.
An algebra homomorphism is a map that preserves the 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 numbers form a group for addition, and the positive real numbers form a group for multiplication. The exponential function

x\mapsto e^{x}

satisfies

e^{x+y}=e^{x}e^{y},

and is thus an homomorphism between these two groups. It is even an isomorphism (see below), as its inverse function, the natural logarithm, satisfies

\ln(xy)=\ln(x)+\ln(y),

and is also a group homomorphism.

Examples

The real numbers are a ring, having both addition and multiplication. The set of all 2 × 2 matrices is also a ring, under matrix addition and matrix multiplication. If we define a function between these rings as follows:

f(r)={\begin{pmatrix}r&0\\0&r\end{pmatrix}}

where r is a real number, then f is a homomorphism of rings, since f preserves both addition:

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)

and multiplication:

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).

For another example, the nonzero complex numbers form a 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, ƒ(z) is the absolute value (or modulus) of the complex number z. Then f is a homomorphism of groups, since it preserves multiplication:

f(z₁ z₂) = |z₁ z₂| = |z₁| |z₂| = f(z₁) f(z₂).

Note that ƒ cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:

|z₁ + z₂| ≠ |z₁| + |z₂|.

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

Special homomorphisms

Several kinds of homomorphisms have a specific name, which is also defined for general morphisms.

Isomorphism

An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.^[1]^: 134 ^[2]^: 28

In the more general context of category theory, an isomorphism is defined as a morphism, which has an 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

f:A\to B

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

g:B\to A

such that

f\circ g=\operatorname {Id} _{B}\qquad {\text{and}}\qquad g\circ f=\operatorname {Id} _{A}.

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

Conversely, if $f:A\to B$ is a bijective homomorphism between algebraic structures, let $g:B\to A$ be the map such that $g (y)$ is the unique element $x$ of $A$ such that $f (x) = y$ . One has $f\circ g=\operatorname {Id} _{B}{\text{ and }}g\circ f=\operatorname {Id} _{A},$ and it remains only to shows that $g$ is an homomorphism. If $*$ is a binary operation of the structure, for every pair $x, y$ of elements of $B$ , one has

g(x*_{B}y)=g(f(g(x))*_{B}f(g(y)))=g(f(g(x)*_{A}g(y)))=g(x)*_{A}g(y),

and $g$ is thus compatible with $*.$ As the proof is similar for any arity, this shows that $g$ is an homomorphism.

This proof does not work for non-algebraic structures. For examples, for topological spaces, a morphism is a continuous map, and the inverse of a bijective continuous map needs not to be continuous. An isomorphism of topological spaces, called homeomorphism or bicontinuous map, is thus a bijective continuous map, whose inverse is also continuous.

Endomorphism

An endomorphism is an homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to the target.^[1]^: 135

The endomorphisms of an algebraic structure, or of an object of a category form a monoid under composition.

The endomorphisms of a vector space or of a module form a ring. In the case of a vector space or a free module of finite dimension, the choice of a 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, which also an isomorphism.^[1]^: 135

The automorphisms of an algebraic structure or of an object of a category form a 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 $\operatorname {GL} _{n}(k)$ is the automorphism group of a vector space of dimension $n$ over a field $k$ .

The automorphism groups of fields were introduced by Évariste Galois for studying the roots of polynomials, and are the basis of Galois theory.

Monomorphism

For algebraic structures, monomorphisms are commonly defined as injective homomorphisms.^[1]^: 134 ^[2]^: 29

In the more general context of category theory, a monomorphism is defined as an homomorphism that is left cancelable.^[3] This means that a (homo)morphism $f:A\to B$ is a monomorphism if, for any pair $g, h$ of morphisms from any other object $C$ to $A$ , then $f\circ g=f\circ h$ implies $g = h$ .

These two definitions of monomorphism are equivalent for all common algebraic structures. More precisely, they are equivalent for fields, for which every homomorphism is a monomorphism, and for 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 sets, magmas, semigroups, monoids, groups, rings, fields, vector spaces and modules.

A split monomorphism is an homomorphism that has a right inverse. That is, an homomorphisms $f\colon A\to B$ is a split homomorphism if there exists an homomorphism $g\colon B\to A$ such that $g\circ f=\operatorname {Id} _{A}.$ 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 is wrong for most common algebraic structures.

Proof of the equivalence of the two definitions of monomorphisms

An injective homomorphism is left cancelable: If $f\circ g=f\circ h,$ one has $f(g(x))=f(h(x))$ for every $x$ in $C$ , the common source of $g$ and $h$ . If $f$ is injective, then $g (x) = h (x)$ , and thus $g = h$ . This proof works not only for algebraic structures, but also for any 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 spaces.

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

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

Existence of a free object on $x$ for a variety (see also Free object § Existence): For building a free object over $x$ , let us consider the set $W$ of the well-formed formulas built up from $x$ 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 (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 classes of $W$ for this relation. It is straightforward to show that the resulting object is a free object on $W$ .

Epimorphism

In algebra, epimorphisms are often defined as surjective homomorphisms.^[1]^: 134^[2]^: 43. On the other hand, in category theory, epimorphisms are defined as right cancelable.^[3] This means that a (homo)morphism $f:A\to B$ is an epimorphism if, for any pair $g, h$ of morphisms from $B$ to any other object $C$ , the equality $g\circ f=h\circ f$ implies $g = h$ .

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 sets, vector spaces, abelian groups and modules. 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.

The most basic example of an epimorphism (category theory meaning), which is not surjective, is the ring inclusion of Z in Q.^[3]^[4] This is also an example of a ring homomorphism which is both a monomorphism and an epimorphism, but not an isomorphism.

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 an homomorphism that has a left inverse. That is, an homomorphisms $f\colon A\to B$ is a split epimorphism if there exists an homomorphism $g\colon B\to A$ such that $f\circ g=\operatorname {Id} _{B}.$ 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 is wrong for most common algebraic structures.

In summary, one has

{\text{split epimorphism}}\implies {\text{epimorphism (surjective)}}\implies {\text{epimorphism (right cancelable)}};

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.

Equivalence of the two definitions of epimorphism

Let $f\colon A\to B$ be an homorphism. We want to prove that if it is not surjective, it is not right cancelable.

In the case of sets, let $b$ be an element of $B$ that not belongs to $f (A)$ , and define $g,h\colon B\to B$ such that $g$ is the identity function, and that $h (x) = x$ for every $x\in B,$ except that $h (b)$ is any other element of $B$ . Clearly $f$ is not right cancelable, as $g \neq h$ and $g\circ f=h\circ f.$

In the case of vector spaces, abelian groups and module, the proof relies on the existence of cokernels and on the fact that the zero maps are homomorphisms: let $C$ be the cockernel of $f$ , and $g\colon B\to C$ be the canonical map, such that $g (f (A)) = 0$ . Let $h=k\circ g$ where $k$ is the zero map from $C$ to $C$ . If $f$ is not surjective, $C \neq 0$ , and thus $g \neq h$ (one is a zero map, while the other is not). Thus $f$ is not cancelable, as $g\circ f=h\circ f$ (both are the zero map from $A$ to $C$ ).

Kernel

Any homomorphism $f : X \to Y$ defines an equivalence relation $~$ on $X$ by $a ~ b$ if and only if $f (a) = f (b)$ . The relation $~$ is called the kernel of $f$ . It is a congruence relation on $X$ . The quotient set $X / ~$ can then be given a structure of the same type, in a natural way, by defining the operations as $[x] * [y] = [x * y]$ . In that case the image of $X$ in $Y$ under the homomorphism $f$ is necessarily isomorphic to $X / ~$ ; this fact is one of the isomorphism theorems.

When the algebraic structure is a group for some operation, the equivalence class $K$ of the identity element of this operation suffices to characterize the equivalence relation. In ths case, the quotient by the equivalence relation is denoted by $X / K$ (usually read as " $X$ mod $K$ "). Also in this case, it is $K$ , rather than $~$ , that is called the kernel of $f$ . 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 groups, vector spaces and modules, but is different and has received a specific name in other cases, such as normal subgroup for kernels of group homomorphisms and ideals for kernels of ring homomorphisms (in the case of non-commutative rings, the kernels are the two-sided ideals).

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^A(a₁,…,a_n)) = F^B(h(a₁),…,h(a_n)) for each n-ary function symbol F in L,
R^A(a₁,…,a_n) implies R^B(h(a₁),…,h(a_n)) 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.^[5]

Formal language theory

Homomorphisms are also used in the study of formal languages^[6] (although within this context, often they are briefly referred to as morphisms^[7]). Given alphabets Σ₁ and Σ₂, a function h : Σ₁^∗ → Σ₂^∗ such that h(uv) = h(u) h(v) for all u and v in Σ₁^∗ is called a homomorphism (or simply morphism) on Σ₁^∗.^{[note 2]} Let e denote the empty word. If h is a homomorphism on Σ₁^∗ and h(x) ≠ e for all x ≠ e in Σ₁^∗, 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^∗ 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.

Notes

^ As it is often the case, but not always, the same symbol for the operation of both $A$ and $B$ was used here.
^ In homomorphisms on formal languages, the ∗ operation is the Kleene star operation. The ⋅ and ∘ are both concatenation, commonly denoted by juxtaposition.

References

^ ^a ^b ^c ^d ^e Birkhoff, Garrett (1967) [1940], Lattice theory, American Mathematical Society Colloquium Publications, vol. 25 (3rd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1025-5, MR 0598630
^ ^a ^b ^c Stanley N. Burris; H.P. Sankappanavar (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
^ ^a ^b ^c Mac Lane, Saunders (1971). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5. Springer-Verlag. Exercise 4 in section I.5. ISBN 0-387-90036-5. Zbl 0232.18001.
^ Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Şerban (2001). Hopf Algebra: An Introduction. Pure and Applied Mathematics. Vol. 235. New York, NY: Marcel Dekker. p. 363. ISBN 0824704819. Zbl 0962.16026.
^ Section 17.4, in Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7
^ Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, ISBN 0-7204-2506-9.
^ 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.

[1] As it is often the case, but not always, the same symbol for the operation of both $A$ and $B$ was used here.

[9] In homomorphisms on formal languages, the ∗ operation is the Kleene star operation. The ⋅ and ∘ are both concatenation, commonly denoted by juxtaposition.

[Birkhoff.1967-2] Birkhoff, Garrett (1967) [1940], Lattice theory, American Mathematical Society Colloquium Publications, vol. 25 (3rd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1025-5, MR 0598630

[Burris.Sankappanavar.2012-3] Stanley N. Burris; H.P. Sankappanavar (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.

[workmath-4] Mac Lane, Saunders (1971). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5. Springer-Verlag. Exercise 4 in section I.5. ISBN 0-387-90036-5. Zbl 0232.18001.

[5] Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Şerban (2001). Hopf Algebra: An Introduction. Pure and Applied Mathematics. Vol. 235. New York, NY: Marcel Dekker. p. 363. ISBN 0824704819. Zbl 0962.16026.

[6] Section 17.4, in Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7

[7] Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, ISBN 0-7204-2506-9.

[8] 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.

[note 1]

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@@ Line 77: / Line 77: @@
 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}}.
-== Types ==
+== Special homomorphisms ==
-Several kinds of homomorphisms have a specific name, which is also defined for general [[morphism]]s. The general definition of the three first kinds of (homo)morphisms is given [[#Category theory|below]].
+Several kinds of homomorphisms have a specific name, which is also defined for general [[morphism]]s.
-* An '''[[isomorphism]]''' is a homomorphism that has an [[inverse function|inverse]]. For structures with an underlying set, this implies that an isomorphism is [[bijective]]. For most{{cn|reason=Isn't that 'all', in both a category-theory and a set-theory setting?|date=December 2016}} algebraic structures a bijective homomorphism is an isomorphism. Thus some authors define isomorphisms as bijective homomorphims.<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}}
-* A '''[[monomorphism]]''' is generally [[injective]]. For most algebraic structures, an injective homomorphism is a monomorphism.{{cn|date=December 2016}} Thus some authors define monomorphisms as injective homomorphims.<ref name="Birkhoff.1967"/>{{rp|134}}<ref name="Burris.Sankappanavar.2012"/>{{rp|29}} A homomorphism that has a [[left inverse function|left inverse]] is a monomorphism, but the converse is not true (for example) for [[module homomorphism]]s.
-* An '''[[epimorphism]]''' is often [[surjective]], but this is not true for [[ring homomorphism]]s. For most algebraic structures, a surjective homomorphism is an epimorphism. For many algebraic structures, including [[group (mathematics)|group]]s, [[vector space]]s and [[module (mathematics)|modules]], a homomorphism is an epimorphism if and only if it is surjective.{{cn|date=December 2016}} Thus some authors define epimorphisms as surjective homomorphims.<ref name="Birkhoff.1967"/>{{rp|134}}<ref name="Burris.Sankappanavar.2012"></ref>{{rp|43}} A homomorphism that has a [[left inverse function|right inverse]] is an epimorphism, but the converse is not true, for example for module homomorphisms.{{cn|date=December 2016}}
-* An '''[[endomorphism]]''' is a homomorphism from a structure to itself.<ref name="Birkhoff.1967"/>{{rp|135}}
-* An '''[[automorphism]]''' is an endomorphism which is also an isomorphism, i.e., an isomorphism from a structure to itself.<ref name="Birkhoff.1967"/>{{rp|135}}
+=== Isomorphism ===
-If there is an isomorphism between two structures, they are completely indistinguishable as far as the structure in question is concerned; in this case, they are said to be ''isomorphic''.
+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.
-The inclusion of [[Integer|'''Z''']] in [[rational number|'''Q''']] is a [[ring homomorphism]], which is not surjective. This is an example of a ring epimorphism, which is not surjective.<ref>{{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><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>  This is also an example of a ring homomorphism which is both a monomorphism and an epimorphism, but not an isomorphism.
+More precisely, if
-===Category theory===
+:<math>f: A\to B</math>
-{{main|Morphism#Some specific morphisms}}
+is a (homo)morphism, it has an inverse if there exists an homorphism
-The general definitions of isomorphism, monomorphisms and epimorphisms belongs to [[category theory]], and are recalled here.
+:<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''}}.
-A morphism {{math|''f'' : ''A'' → ''B''}} is called:
-* a '''monomorphism''' if {{math|1=''f'' ∘ ''g''<sub>1</sub> = ''f'' ∘ ''g''<sub>2</sub>}} implies {{math|1=''g''<sub>1</sub> = ''g''<sub>2</sub>}} for all morphisms {{math|''g''<sub>1</sub>, ''g''<sub>2</sub>: ''X'' → ''A''}}, where "∘" denotes (homo)morphism composition (a sufficient condition for this is {{math|''f''}} having a left inverse).
+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 shows that {{math|''g''}} is an homomorphism. If <math>*</math> is a binary operation of the structure, for every pair {{math|''x'', ''y''}} of elements of {{math|''B''}}, one has
-* an '''epimorphism''' if {{math|1=''g''<sub>1</sub> ∘ ''f'' = ''g''<sub>2</sub> ∘ ''f''}} implies {{math|1=''g''<sub>1</sub> = ''g''<sub>2</sub>}} for all morphisms {{math|''g''<sub>1</sub>, ''g''<sub>2</sub>: ''B'' → ''X''}} (a sufficient condition for this is {{math|''f''}} having a right inverse).
+:<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>
-* an '''isomorphism''' if there exists a morphism {{math|''g'': ''B'' → ''A''}} such that {{math|1=''f'' ∘ ''g'' = 1<sub>''B''</sub>}} and {{math|1=''g'' ∘ ''f'' = 1<sub>''A''</sub>}}, where {{math|"1<sub>''X''</sub>"}} denotes the identity morphism on the object {{math|''X''}}.
+and {{math|''g''}} is thus compatible with <math>*.</math> As the proof is similar for any [[arity]], this shows that {{math|''g''}} is an 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 needs not to be 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 an 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, which 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 an homomorphism that is '''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 an homomorphism that has a [[inverse function|right inverse]]. That is, an homomorphisms <math>f\colon A \to B</math> is a split homomorphism if there exists an 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 is wrong 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 homomophism 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>\Bbb{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 homomophism, 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 an 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''}}, let us 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"></ref>{{rp|43}}. On the other hand, in [[category theory]], [[epimorphism]]s are defined as '''right cancelable'''.<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 and [[module (mathematics)|modules]]. 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.
+The most basic example of an epimorphism (category theory meaning), which is not surjective, is the [[ring (mathematics)|ring]] inclusion of [[Integer|'''Z''']] in [[rational number|'''Q''']].<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>  This is also an example of a ring homomorphism which is both a monomorphism and an epimorphism, but not an isomorphism.
+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 an homomorphism that has a [[inverse function|left inverse]]. That is, an homomorphisms <math>f\colon A \to B</math> is a split epimorphism if there exists an 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 is wrong 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 an homorphism. 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 module, 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 cockernel 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= k\circ g</math> where {{math|''k''}} is the zero map from {{math|''C''}} to {{math|''C''}}. 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 {{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, in which case 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]]).
+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, in a natural way, by defining the operations as {{math|1=[''x''] ∗ [''y''] =  [''x'' ∗ ''y'']}}. 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 ths 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 ==