Homomorphism

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: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
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 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 structures of the same type (that is of the same name), that preserves the operations of the structures. This means a map formula_1 between two sets formula_2, formula_3 equipped with the same structure such that, if formula_4 is an operation of the structure (supposed here, for simplification, to be a binary operation), then
for every pair formula_6, formula_7 of elements of formula_2. One says often that formula_9 preserves the operation or is compatible with the operation.
Formally, a map formula_10 preserves an operation formula_11 of arity "k", defined on both formula_2 and formula_3 if 
for all elements formula_15 in formula_2.
The operations that must be preserved by a homomorphism include 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:
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 
satisfies
and is thus a homomorphism between these two groups. It is even an isomorphism (see below), as its inverse function, the natural logarithm, satisfies 
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:
where is a real number, then is a homomorphism of rings, since preserves both addition:
and multiplication:
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 formula_9 from the nonzero complex numbers to the nonzero real numbers by
That is, formula_9 is the absolute value (or modulus) of the complex number formula_26. Then formula_9 is a homomorphism of groups, since it preserves multiplication:
Note that cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition:
As another example, the diagram shows a monoid homomorphism formula_9 from the monoid formula_31 to the monoid formula_32. Due to the different names of corresponding operations, the structure preservation properties satisfied by formula_9 amount to formula_34 and formula_35.
A composition algebra formula_2 over a field formula_37 has a quadratic form, called a "norm", formula_38, which is a group homomorphism from the multiplicative group of formula_2 to the multiplicative group of formula_37.
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. 
In the more general context of category theory, an isomorphism is defined as a morphism that 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 
is a (homo)morphism, it has an inverse if there exists a homomorphism
such that
If formula_2 and formula_3 have underlying sets, and formula_1 has an inverse formula_47, then formula_9 is bijective. In fact, formula_9 is injective, as formula_50 implies formula_51, and formula_9 is surjective, as, for any formula_6 in formula_3, one has formula_55, and formula_6 is the image of an element of formula_2.
Conversely, if formula_1 is a bijective homomorphism between algebraic structures, let formula_59 be the map such that formula_60 is the unique element formula_6 of formula_2 such that formula_63. One has formula_64 and it remains only to show that is a homomorphism. If formula_65 is a binary operation of the structure, for every pair formula_6, formula_7 of elements of formula_3, one has
and formula_47 is thus compatible with formula_71 As the proof is similar for any arity, this shows that formula_47 is a 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 is not necessarily 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 a homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to the target.
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 that is also an isomorphism.
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 formula_73 is the automorphism group of a vector space of dimension formula_74 over a field formula_75.
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. 
In the more general context of category theory, a monomorphism is defined as a morphism that is left cancelable. This means that a (homo)morphism formula_76 is a monomorphism if, for any pair formula_47, formula_78 of morphisms from any other object formula_79 to formula_2, then formula_81 implies formula_82.
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 a homomorphism that has a left inverse and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism formula_83 is a split monomorphism if there exists a homomorphism formula_84 such that formula_85 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.
"An injective homomorphism is left cancelable": If formula_86 one has formula_87 for every formula_6 in formula_79, the common source of formula_47 and formula_78. If formula_9 is injective, then formula_93, and thus formula_82. 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 homomorphism is injective, it is useful to consider a "free object on formula_6". Given a variety of algebraic structures a free object on formula_6 is a pair consisting of an algebraic structure formula_97 of this variety and an element formula_6 of formula_97 satisfying the following universal property: for every structure formula_100 of the variety, and every element formula_101 of formula_100, there is a unique homomorphism formula_103 such that formula_104. For example, for sets, the free object on formula_6 is simply formula_106; for semigroups, the free object on formula_6 is formula_108 which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for monoids, the free object on formula_6 is formula_110 which, as, a monoid, is isomorphic to the additive monoid of the nonnegative integers; for groups, the free object on formula_6 is the infinite cyclic group formula_112 which, as, a group, is isomorphic to the additive group of the integers; for rings, the free object on formula_6} is the polynomial ring formula_114 for vector spaces or modules, the free object on formula_6 is the vector space or free module that has formula_6 as a basis.
"If a free object over formula_6 exists, then every left cancelable homomorphism is injective": let formula_83 be a left cancelable homomorphism, and formula_119 and formula_120 be two elements of formula_2 such formula_122. By definition of the free object formula_37, there exist homomorphisms formula_47 and formula_78 from formula_37 to formula_2 such that formula_128 and formula_129. As formula_130, one has formula_131 by the uniqueness in the definition of a universal property. As formula_9 is left cancelable, one has formula_82, and thus formula_134. Therefore, formula_9 is injective.
"Existence of a free object on formula_6 for a variety" (see also ): For building a free object over formula_6, consider the set formula_138 of the well-formed formulas built up from formula_6 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 formula_138 for this relation. It is straightforward to show that the resulting object is a free object on formula_138.
Epimorphism.
In algebra, epimorphisms are often defined as surjective homomorphisms. On the other hand, in category theory, epimorphisms are defined as right cancelable morphisms. This means that a (homo)morphism formula_1 is an epimorphism if, for any pair formula_47, formula_78 of morphisms from formula_3 to any other object formula_79, the equality formula_147 implies formula_82.
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, modules (see below for a proof), and groups. The importance of these structures in all mathematics, and specially in linear algebra and homological algebra, may explain the coexistence of two non-equivalent definitions.
Algebraic structures for which there exist non-surjective epimorphisms include semigroups and rings. The most basic example is the inclusion of integers into rational numbers, which is an homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism.
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 right inverse and thus it is itself a left inverse of that other homomorphism. That is, a homomorphism formula_83 is a split epimorphism if there exists a homomorphism formula_84 such that formula_151 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 
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.
Let formula_83 be a homomorphism. We want to prove that if it is not surjective, it is not right cancelable.
In the case of sets, let formula_120 be an element of formula_3 that not belongs to formula_156, and define formula_157 such that formula_47 is the identity function, and that formula_159 for every formula_160 except that formula_161 is any other element of formula_3. Clearly formula_9 is not right cancelable, as formula_164 and formula_165
In the case of vector spaces, abelian groups and modules, the proof relies on the existence of cokernels and on the fact that the zero maps are homomorphisms: let formula_79 be the cokernel of formula_9, and formula_168 be the canonical map, such that formula_169. Let formula_170 be the zero map. If formula_9 is not surjective, formula_172, and thus formula_164 (one is a zero map, while the other is not). Thus formula_9 is not cancelable, as formula_147 (both are the zero map from formula_2 to formula_79).
Kernel.
Any homomorphism formula_178 defines an equivalence relation formula_179 on formula_180 by formula_181 if and only if formula_122. The relation formula_179 is called the kernel of formula_9. It is a congruence relation on formula_180. The quotient set formula_186 can then be given a structure of the same type as formula_180, in a natural way, by defining the operations of the quotient set by formula_188, for each operation formula_189 of formula_180. In that case the image of formula_180 in formula_192 under the homomorphism formula_9 is necessarily isomorphic to formula_194; this fact is one of the isomorphism theorems.
When the algebraic structure is a group for some operation, the equivalence class formula_195 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 formula_196 (usually read as "formula_180 mod formula_195"). Also in this case, it is formula_195, rather than formula_179, that is called the kernel of formula_9. 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
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.
Formal language theory.
Homomorphisms are also used in the study of formal languages and are often briefly referred to as morphisms. Given alphabets Σ1 and Σ2, a function such that for all "u" and "v" in Σ1∗ is called a "homomorphism" on Σ1∗. If "h" is a homomorphism on Σ1∗ and ε denotes the empty string, then "h" is called an "ε-free homomorphism" when for all in Σ1∗.
The set Σ∗ of words formed from the alphabet Σ may be thought of as the free monoid generated by Σ. Here the monoid operation is concatenation and the identity element is the empty word. From this perspective, a language homormorphism is precisely a monoid homomorphism.