Structure-preserving map between two algebraic structures of the same type

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". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same".[1] The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).[2]

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.

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.

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

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

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

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:

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

An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.[3]: 134  [4]: 28 

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.

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.

An endomorphism is a homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to the target.[3]: 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.

An automorphism is an endomorphism that is also an isomorphism.[3]: 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.

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

For algebraic structures, monomorphisms are commonly defined as injective homomorphisms.[3]: 134  [4]: 29 

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 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.[6] 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.[5][7]

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.

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.

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.[8]

Homomorphisms are also used in the study of formal languages[9] and are often briefly referred to as morphisms.[10] 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 on Σ1.[note 2] If h is a homomorphism on Σ1 and ε denotes the empty string, then h is called an ε-free homomorphism when h(x) ≠ ε for all xε 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.[note 3]