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.
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 between two sets, equipped with the same structure such that, if is an operation of the structure (supposed here, for simplification, to be a binary operation), then
for every pair , of elements of .[note 1] One says often that preserves the operation or is compatible with the operation.
Formally, a map preserves an operation of arityk, defined on both and if
for all elements in .
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.
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.
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
where r is a real number, then f is a homomorphism of rings, since f preserves both addition:
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 from the nonzero complex numbers to the nonzero real numbers by
That is, is the absolute value (or modulus) of the complex number . Then is a homomorphism of groups, since it preserves multiplication:
Note that f 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 from the monoid to the monoid . Due to the different names of corresponding operations, the structure preservation properties satisfied by amount to and .
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
If and have underlying sets, and has an inverse , then is bijective. In fact, is injective, as implies , and is surjective, as, for any in , one has , and is the image of an element of .
Conversely, if is a bijective homomorphism between algebraic structures, let be the map such that is the unique element of such that . One has and it remains only to show that g is a homomorphism. If is a binary operation of the structure, for every pair , of elements of , one has
and is thus compatible with As the proof is similar for any arity, this shows that 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.
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 is a monomorphism if, for any pair , of morphisms from any other object to , then implies .
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).
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 is a split homomorphism if there exists a homomorphism such that A split monomorphism is always a monomorphism, for both meanings of monomorphism. For sets and vector spaces, every monomorphism is a split homomorphism, but this property does not hold for most common algebraic structures.
Proof of the equivalence of the two definitions of monomorphisms
An injective homomorphism is left cancelable: If one has for every in , the common source of and . If is injective, then , and thus . 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 . Given a variety of algebraic structures a free object on is a pair consisting of an algebraic structure of this variety and an element of satisfying the following universal property: for every structure of the variety, and every element of , there is a unique homomorphism such that . For example, for sets, the free object on is simply ; for semigroups, the free object on is which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for monoids, the free object on is which, as, a monoid, is isomorphic to the additive monoid of the nonnegative integers; for groups, the free object on is the infinite cyclic group which, as, a group, is isomorphic to the additive group of the integers; for rings, the free object on } is the polynomial ring for vector spaces or modules, the free object on is the vector space or free module that has as a basis.
If a free object over exists, then every left cancelable homomorphism is injective: let be a left cancelable homomorphism, and and be two elements of such . By definition of the free object , there exist homomorphisms and from to such that and . As , one has by the uniqueness in the definition of a universal property. As is left cancelable, one has , and thus . Therefore, is injective.
Existence of a free object on for a variety (see also Free object § Existence): For building a free object over , consider the set of the well-formed formulas built up from 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 for this relation. It is straightforward to show that the resulting object is a free object on .
In algebra, epimorphisms are often defined as surjective homomorphisms.:134:43 On the other hand, in category theory, epimorphisms are defined as right cancelablemorphisms. This means that a (homo)morphism is an epimorphism if, for any pair , of morphisms from to any other object , the equality implies .
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 is a split epimorphism if there exists a homomorphism such that 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.
Equivalence of the two definitions of epimorphism
Let be a homomorphism. We want to prove that if it is not surjective, it is not right cancelable.
In the case of sets, let be an element of that not belongs to , and define such that is the identity function, and that for every except that is any other element of . Clearly is not right cancelable, as and
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 be the cokernel of , and be the canonical map, such that . Let be the zero map. If is not surjective, , and thus (one is a zero map, while the other is not). Thus is not cancelable, as (both are the zero map from to ).
Any homomorphism defines an equivalence relation on by if and only if . The relation is called the kernel of . It is a congruence relation on . The quotient set can then be given a structure of the same type as , in a natural way, by defining the operations of the quotient set by , for each operation of . In that case the image of in under the homomorphism is necessarily isomorphic to ; this fact is one of the isomorphism theorems.
When the algebraic structure is a group for some operation, the equivalence class 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 (usually read as "mod"). Also in this case, it is , rather than , that is called the kernel of . 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).
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(FA(a1,...,an)) = FB(h(a1),...,h(an)) for each n-ary function symbol F in L,
RA(a1,...,an) implies RB(h(a1),...,h(an)) 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.
Formal language theory
Homomorphisms are also used in the study of formal languages and are often briefly referred to as morphisms. Given alphabets ?1 and ?2, a function such that for all u and v in ?1* is called a homomorphism on ?1*.[note 2] If h is a homomorphism on ?1* and e denotes the empty word, then h is called an e-free homomorphism when for all in ?1*.
The set ?* of words formed from the alphabet ? may be thought of as the free monoid generated by ?. 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.[note 3]
^As it is often the case, but not always, the same symbol for the operation of both and was used here.
^The * denotes the Kleene star operation, while ?* denotes the set of words formed from the alphabet ?, including the empty word. Juxtaposition of terms denotes concatenation. For example, h(u) h(v) denotes the concatenation of h(u) with h(v).
^We are assured that a language homomorphism h maps the empty word e to the empty word. Since h(e) = h(ee) = h(e)h(e), the number w of characters in h(e) equals the number 2w of characters in h(e)h(e). Hence w = 0 and h(e) has null length.
Ritter, Ernst (1892). "Die eindeutigen automorphen Formen vom Geschlecht Null, eine Revision und Erweiterung der Poincaré'schen Sätze" [The unique automorphic forms of genus zero, a revision and extension of Poincaré's theorem]. Mathematische Annalen (in German). 41: 1-82. doi:10.1007/BF01443449. S2CID121524108. From footnote on 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, ... " (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", ... )
Fricke, Robert (1892). "Ueber den arithmetischen Charakter der zu den Verzweigungen (2,3,7) und (2,4,7) gehörenden Dreiecksfunctionen" [On the arithmetic character of the triangle functions belonging to the branch points (2,3,7) and (2,4,7)]. Mathematische Annalen (in German). 41: 443-468. doi:10.1007/BF01443421. S2CID120022176. From p. 466:"Hierdurch ist, wie man sofort überblickt, eine homomorphe*) Beziehung der Gruppe ?(63) auf die Gruppe der mod. n incongruenten Substitutionen mit rationalen ganzen Coefficienten der Determinante 1 begründet." (Thus, as one immediately sees, a homomorphic relation of the group ?(63) is based on the group of modulo n incongruent substitutions with rational whole coefficients of the determinant 1.) From footnote on p. 466: "*) 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"." (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".)