Homomorphism lemma
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In algebra, a homomorphism is a structure-preserving map between two
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s of the same type (such as two groups, two
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s, 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 subject of linear algebra. The concept of homomorphism has been generalized, under the name of
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
, 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
. A homomorphism may also be an isomorphism, an endomorphism, an
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
, 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 structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s of the same type (that is of the same name), that preserves the
operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
of the structures. This means a map f: A \to B between two sets A, B equipped with the same structure such that, if \cdot is an operation of the structure (supposed here, for simplification, to be a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
), then :f(x\cdot y)=f(x)\cdot f(y) for every pair x, y of elements of A.As it is often the case, but not always, the same symbol for the operation of both A and B was used here. One says often that f preserves the operation or is compatible with the operation. Formally, a map f: A\to B preserves an operation \mu of
arity Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
''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. 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: * 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
. 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 spaces; 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 Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, is defined similarly. * An
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF(x) ...
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^ = e^xe^y, and is thus a homomorphism between these two groups. It is even an isomorphism (see below), as its inverse function, the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, satisfies :\ln(xy)=\ln(x)+\ln(y), and is also a group homomorphism.


Examples

The real numbers are a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, 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 r & 0 \\ 0 & r \end where is a real number, then is a homomorphism of rings, since preserves both addition: :f(r+s) = \begin r+s & 0 \\ 0 & r+s \end = \begin r & 0 \\ 0 & r \end + \begin s & 0 \\ 0 & s \end = f(r) + f(s) and multiplication: :f(rs) = \begin rs & 0 \\ 0 & rs \end = \begin r & 0 \\ 0 & r \end \begin s & 0 \\ 0 & s \end = 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, f is the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
(or modulus) of the complex number z. Then f is a homomorphism of groups, since it preserves multiplication: :f(z_1 z_2) = , z_1 z_2, = , z_1, , z_2, = f(z_1) f(z_2). Note that cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition: :, z_1 + z_2, \ne , z_1, + , z_2, . As another example, the diagram shows a monoid homomorphism f from the monoid (\mathbb, +, 0) to the monoid (\mathbb, \times, 1). Due to the different names of corresponding operations, the structure preservation properties satisfied by f amount to f(x+y) = f(x) \times f(y) and f(0) = 1. A composition algebra A over a field F has a
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
, called a ''norm'', N: A \to F, which is a group homomorphism from the multiplicative group of A to the multiplicative group of F.


Special homomorphisms

Several kinds of homomorphisms have a specific name, which is also defined for general
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s.


Isomorphism

An isomorphism between
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s of the same type is commonly defined as a bijective homomorphism. In the more general context of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, an isomorphism is defined as a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
that has an
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
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 a homomorphism :g: B\to A such that :f\circ g = \operatorname_B \qquad \text \qquad g\circ f = \operatorname_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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, as f(x) = f(y) implies x = g(f(x)) = g(f(y)) = y, and f is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, 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_B \text g \circ f = \operatorname_A, and it remains only to show that is a 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))*_Bf(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 Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
, this shows that g 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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
equals the codomain, or, more generally, a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
whose source is equal to its 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 Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
form a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
. In the case of a vector space or a
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
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 In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
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 \operatorname_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 A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of polynomials, and are the basis of Galois theory.


Monomorphism

For algebraic structures, monomorphisms are commonly defined as
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
homomorphisms. In the more general context of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a monomorphism is defined as a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
that is left cancelable. 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 (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 sets,
magmas Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
, semigroups, monoids,
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
, rings, fields, vector spaces and
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
. A
split monomorphism In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y is t ...
is a homomorphism that has a left inverse and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism f\colon A \to B is a split monomorphism if there exists a homomorphism g\colon B \to A such that g \circ f = \operatorname_A. 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 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 homomorphism 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 \; for semigroups, the free object on x is \, which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for monoids, the free object on x is \, 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 \, 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 for vector spaces or
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, 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 homomorphism, 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 a 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 ): For building a free object over x, 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each 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 In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of W for this relation. It is straightforward to show that the resulting object is a free object on x.


Epimorphism

In algebra, epimorphisms are often defined as
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
homomorphisms. On the other hand, in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, epimorphisms are defined as right cancelable
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s. 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,
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
(see below for a proof), and
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
. The importance of these structures in all mathematics, and specially in linear algebra and
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, 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 a 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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, this may explain why in these areas, the definition of epimorphisms as right cancelable homomorphisms is generally preferred. A
split epimorphism In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y ...
is a homomorphism that has a right inverse and thus it is itself a left inverse of that other homomorphism. That is, a homomorphism f\colon A \to B is a split epimorphism if there exists a homomorphism g\colon B \to A such that f\circ g = \operatorname_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 does not hold for most common algebraic structures. In summary, one has :\text \implies \text\implies \text ; 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 f\colon A \to B be a homomorphism. 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 modules, the proof relies on the existence of cokernels and on the fact that the zero maps are homomorphisms: let C be the cokernel of f, and g\colon B \to C be the canonical map, such that g(f(A)) = 0. Let h\colon B\to C be the zero map. 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
\sim on X by a \sim b if and only if f(a) = f(b). The relation \sim is called the kernel of f. It is a
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
on X. The quotient set X/ can then be given a structure of the same type as X, in a natural way, by defining the operations of the quotient set by \ast = \ast y/math>, for each operation \ast of X. In that case the image of X in Y under the homomorphism f is necessarily
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to X/\!\sim; this fact is one of the isomorphism theorems. When the algebraic structure is a group for some operation, the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
K 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 X/K (usually read as "X
mod Mod, MOD or mods may refer to: Places * Modesto City–County Airport, Stanislaus County, California, US Arts, entertainment, and media Music * Mods (band), a Norwegian rock band * M.O.D. (Method of Destruction), a band from New York City, US ...
K"). Also in this case, it is K, rather than \sim, 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 Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, but is different and has received a specific name in other cases, such as
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
for kernels of group homomorphisms and
ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
for kernels of ring homomorphisms (in the case of non-commutative rings, the kernels are the
two-sided ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
s).


Relational structures

In
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, 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''(''a''1,…,''a''''n'')) = ''F''''B''(''h''(''a''1),…,''h''(''a''''n'')) for each ''n''-ary function symbol ''F'' in ''L'', * ''R''''A''(''a''1,…,''a''''n'') implies ''R''''B''(''h''(''a''1),…,''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.


Formal language theory

Homomorphisms are also used in the study of formal languages and are often briefly referred to as ''morphisms''. Given alphabets \Sigma_1 and \Sigma_2, a function h \colon \Sigma_1^* \to \Sigma_2^* such that h(uv) = h(u) h(v) for all u,v \in \Sigma_1 is called a ''homomorphism'' on \Sigma_1^*.The ∗ denotes the Kleene star operation, while Σ∗ denotes the set of words formed from the alphabet Σ, including the empty word. Juxtaposition of terms denotes concatenation. For example, ''h''(''u'') ''h''(''v'') denotes the concatenation of ''h''(''u'') with ''h''(''v''). If h is a homomorphism on \Sigma_1^* and \varepsilon denotes the empty string, then h is called an \varepsilon''-free homomorphism'' when h(x) \neq \varepsilon for all x \neq \varepsilon in \Sigma_1^*. A homomorphism h \colon \Sigma_1^* \to \Sigma_2^* on \Sigma_1^* that satisfies , h(a), = k for all a \in \Sigma_1 is called a k''-uniform'' homomorphism. p. 287 If , h(a), = 1 for all a \in \Sigma_1 (that is, h is 1-uniform), then h is also called a ''coding'' or a ''projection''. The set \Sigma^* of words formed from the alphabet \Sigma may be thought of as the free monoid generated by \Sigma. Here the monoid operation is concatenation and the identity element is the empty word. From this perspective, a language homomorphism is precisely a monoid homomorphism.We are assured that a language homomorphism ''h'' maps the empty word ''ε'' to the empty word. Since ''h''(''ε'') = ''h''(''εε'') = ''h''(''ε'')''h''(''ε''), the number ''w'' of characters in ''h''(''ε'') equals the number 2''w'' of characters in ''h''(''ε'')''h''(''ε''). Hence ''w'' = 0 and ''h''(''ε'') has null length.


See also

* Diffeomorphism * Homomorphic encryption *
Homomorphic secret sharing In cryptography, homomorphic secret sharing is a type of secret sharing algorithm in which the secret is encrypted via homomorphic encryption. A homomorphism is a transformation from one algebraic structure into another of the same type so that th ...
– a simplistic decentralized voting protocol *
Morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
* Quasimorphism


Notes


Citations


References

*Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". ''Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26–29, 2006''. Oscar H. Ibarra, Zhe Dang. Springer-Verlag. pp. 280–291. ISBN  3-540-35428-X. * * * {{Authority control Morphisms