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algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, a homomorphism is a structure-preserving
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
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
group 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 ...
s, 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 space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s). The word ''homomorphism'' comes from the
Ancient Greek language Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek Dark ...
: () 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 Christian Felix Klein (; 25 April 1849 â€“ 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
(1849–1925). Homomorphisms of vector spaces are also called
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s, and their study is the subject of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
. 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 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 ...
, an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
, 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 of the structures. This means a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
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 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 ...
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 In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
is a map between
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s that preserves the semigroup operation. * A
monoid homomorphism In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
is a map between
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
s 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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
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 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 In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
is a map between
rings 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 ...
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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
is a homomorphism of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s; that is, a group homomorphism between vector spaces that preserves the abelian group structure and
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
. * A
module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ''R' ...
, 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 Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s form a group for addition, and the positive real numbers form a group for multiplication. The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
: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 In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...
, 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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is also a ring, under
matrix addition In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Kronec ...
and
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
. 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s form a
group 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 ...
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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
, 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 In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
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 In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution c ...
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 In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to ...
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 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 ...
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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 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 space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s, a morphism is a
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
, and the inverse of a bijective continuous map is not necessarily continuous. An isomorphism of topological spaces, called
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
or bicontinuous map, is thus a bijective continuous map, whose inverse is also continuous.


Endomorphism

An
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
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 In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
, 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 Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
form a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
under composition. The endomorphisms of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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 In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, the choice of a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
induces a
ring isomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
between the ring of endomorphisms and the ring of
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
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 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 ...
under composition, which is called the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the structure. Many groups that have received a name are automorphism groups of some algebraic structure. For example, the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
\operatorname_n(k) is the automorphism group of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of dimension n over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
k. The automorphism groups of
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s were introduced by
Évariste Galois Évariste Galois (; ; 25 October 1811 â€“ 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
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
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, and are the basis of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.


Monomorphism

For algebraic structures,
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
s 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 Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
, for which every homomorphism is a monomorphism, and for
varieties Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
of
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
, 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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
is defined either as a
unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on ...
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 ...
,
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s,
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
s,
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 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 ...
,
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s 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 Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s. For proving that, conversely, a left cancelable homomorphism is injective, it is useful to consider a ''
free object In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between ele ...
on x''. Given a
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: 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
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s, the free object on x is \, which, as, a semigroup, is isomorphic to the additive semigroup of the positive integers; for
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
s, the free object on x is \, which, as, a monoid, is isomorphic to the additive monoid of the nonnegative integers; for
group 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 ...
s, the free object on x is the
infinite cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
\, which, as, a group, is isomorphic to the additive group of the integers; for
rings 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 ...
, the free object on x is the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
\mathbb for
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s 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 Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
'' (see also ): For building a free object over x, consider the set W of the
well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be ...
s 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 Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, 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 ...
,
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
s 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 space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s,
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s,
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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
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
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s and
rings 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 ...
. The most basic example is the inclusion of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s into
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, 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 In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \ ...
by a multiplicative set. Every localization is a ring epimorphism, which is not, in general, surjective. As localizations are fundamental in
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
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 Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
, 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
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the name: ...
s and on the fact that the
zero map 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
s 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 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 ...
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 theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...
s. When the algebraic structure is a
group 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 ...
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 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 ...
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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
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 group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s 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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...
and ideals for kernels of
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
s (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 In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent verti ...
.


Formal language theory

Homomorphisms are also used in the study of
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symb ...
s 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 In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics, it is more commonly known as the free monoid c ...
operation, while Σ∗ denotes the set of words formed from the alphabet Σ, including the empty word. Juxtaposition of terms denotes
concatenation In formal language, formal language theory and computer programming, string concatenation is the operation of joining character string (computer science), character strings wikt:end-to-end, end-to-end. For example, the concatenation of "sno ...
. 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 In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero eleme ...
generated by \Sigma. Here the monoid operation is
concatenation In formal language, formal language theory and computer programming, string concatenation is the operation of joining character string (computer science), character strings wikt:end-to-end, end-to-end. For example, the concatenation of "sno ...
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 In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
*
Homomorphic encryption Homomorphic encryption is a form of encryption that permits users to perform computations on its encrypted data without first decrypting it. These resulting computations are left in an encrypted form which, when decrypted, result in an identical ...
* Homomorphic secret sharing – 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 In mathematics, given a group G, a quasimorphism (or quasi-morphism) is a function f:G\to\mathbb which is additive up to bounded error, i.e. there exists a constant D\geq 0 such that , f(gh)-f(g)-f(h), \leq D for all g, h\in G. The least posit ...


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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
. pp. 280–291.
ISBN The International Standard Book Number (ISBN) is a numeric commercial book identifier that is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency. An ISBN is assigned to each separate edition and ...
  3-540-35428-X. * * * {{Authority control Morphisms