homomorphism

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In algebra, 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. Althou ...
between two algebraic structures of the same type (such as two groups, two rings, or two
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
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 ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic per ...
: () 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 – 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 grou ...
(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 mapping V \to W between two vector spaces that pre ...
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, morphism ...
, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of
category theory 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, cat ...
. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.

Definition

A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures. This means a
map 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. Althou ...
$f: A \to B$ 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 ope ...
), then :$f\left(x\cdot y\right)=f\left(x\right)\cdot f\left(y\right)$ 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 ''k'', defined on both $A$ and $B$ if :$f\left(\mu_A\left(a_1, \ldots, a_k\right)\right) = \mu_B\left(f\left(a_1\right), \ldots, f\left(a_k\right)\right),$ 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 s ...
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 a ...
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 is a map between groups that preserves the group operation. This implies that the group homomorphism maps the identity element of the first group to the identity element of the second group, and maps the inverse of an element of the first group to the inverse of the image of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism. * A ring homomorphism is a map between rings that preserves the ring addition, the ring multiplication, and the multiplicative identity. Whether the multiplicative identity is to be preserved depends upon the definition of ''ring'' in use. If the multiplicative identity is not preserved, one has a rng homomorphism. * 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 mapping V \to W between two vector spaces that pre ...
is a homomorphism of
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s; that is, a group homomorphism between vector spaces that preserves the abelian group structure and scalar multiplication. * 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 '' ...
, also called a linear map between modules, is defined similarly. * An algebra homomorphism 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 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. Ever ...
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\ ...
, the natural logarithm, satisfies :$\ln\left(xy\right)=\ln\left(x\right)+\ln\left(y\right),$ 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. Ever ...
s are a ring, having both addition and multiplication. The set of all 2×2 matrices 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 Kroneck ...
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 ...
. If we define a function between these rings as follows: :$f\left(r\right) = \begin r & 0 \\ 0 & r \end$ where is a real number, then is a homomorphism of rings, since preserves both addition: :$f\left(r+s\right) = \begin r+s & 0 \\ 0 & r+s \end = \begin r & 0 \\ 0 & r \end + \begin s & 0 \\ 0 & s \end = f\left(r\right) + f\left(s\right)$ and multiplication: :$f\left(rs\right) = \begin rs & 0 \\ 0 & rs \end = \begin r & 0 \\ 0 & r \end \begin s & 0 \\ 0 & s \end = f\left(r\right)\,f\left(s\right).$ 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 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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
, which is required for elements of a group.) Define a function $f$ from the nonzero complex numbers to the nonzero real numbers by :$f\left(z\right) = , z, .$ That is, $f$ is the absolute value (or modulus) of the complex number $z$. Then $f$ is a homomorphism of groups, since it preserves multiplication: :$f\left(z_1 z_2\right) = , z_1 z_2, = , z_1, , z_2, = f\left(z_1\right) f\left(z_2\right).$ 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 a ...
homomorphism $f$ from the monoid $\left(\mathbb, +, 0\right)$ to the monoid $\left(\mathbb, \times, 1\right)$. Due to the different names of corresponding operations, the structure preservation properties satisfied by $f$ amount to $f\left(x+y\right) = f\left(x\right) \times f\left(y\right)$ and $f\left(0\right) = 1$. 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 involuti ...
$A$ 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 t ...
, 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, morphism ...
s.

Isomorphism

An isomorphism between algebraic structures 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, cat ...
, 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, morphism ...
that has an inverse that is also a morphism. In the specific case of algebraic structures, the two definitions are equivalent, although they may differ for non-algebraic structures, which have an underlying set. More precisely, if :$f: A\to B$ 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 contrapositi ...
, as $f\left(x\right) = f\left(y\right)$ implies $x = g\left(f\left(x\right)\right) = g\left(f\left(y\right)\right) = y$, 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 ...
, as, for any $x$ in $B$, one has $x = f\left(g\left(x\right)\right)$, 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 $g\left(y\right)$ is the unique element $x$ of $A$ such that $f\left(x\right) = y$. 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 :$g\left(x*_B y\right) = g\left(f\left(g\left(x\right)\right)*_Bf\left(g\left(y\right)\right)\right) = g\left(f\left(g\left(x\right)*_A g\left(y\right)\right)\right) = g\left(x\right)*_A g\left(y\right),$ and $g$ is thus compatible with $*.$ As the proof is similar for any arity, 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 poi ...
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 valu ...
, 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 isomo ...
or bicontinuous map, is thus a bijective continuous map, whose inverse is also continuous.

Endomorphism

An endomorphism is a homomorphism whose domain equals the codomain, or, more generally, a
morphism 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, morphism ...
whose source is equal to its target. The endomorphisms of an algebraic structure, or of an object of a category 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 a ...
under composition. The endomorphisms of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
or of a module form a ring. In the case of a vector space or a free module of finite
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
, the choice of a basis 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 preserv ...
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 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 $\operatorname_n\left(k\right)$ is the automorphism group of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
of dimension $n$ over a field $k$. The automorphism groups of fields were introduced by Évariste Galois for studying the roots 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 examp ...
s, and are the basis of Galois theory.

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 monomorphi ...
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 contrapositi ...
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, cat ...
, 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, morphism ...
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 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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
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,
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 a ...
s, groups, rings,
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 (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s and modules. A split monomorphism is a homomorphism that has a left inverse and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism $f\colon A \to B$ 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 $f\left(g\left(x\right)\right)=f\left(h\left(x\right)\right)$ for every $x$ in $C$, the common source of $g$ and $h$. If $f$ is injective, then $g\left(x\right) = h\left(x\right)$, 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 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 poi ...
s. 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 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 fr ...
: for every structure $S$ of the variety, and every element $s$ of $S$, there is a unique homomorphism $f: L\to S$ such that $f\left(x\right) = s$. 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 a ...
s, 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 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, 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 variab ...
for
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s or modules, 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 $f\left(a\right) = f\left(b\right)$. By definition of the free object $F$, there exist homomorphisms $g$ and $h$ from $F$ to $A$ such that $g\left(x\right) = a$ and $h\left(x\right) = b$. As $f\left(g\left(x\right)\right) = f\left(h\left(x\right)\right)$, 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 relatio ...
, if the identities are not subject to conditions, that is if one works with a variety. Then the operations of the variety are well defined on the set of equivalence classes of $W$ 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 ...
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, cat ...
, 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, morphism ...
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 (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s, abelian groups, modules (see below for a proof), and groups. 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 in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
, 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. 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 languag ...
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 ratio ...
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 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. Promine ...
and algebraic geometry, 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 ...
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 $f\left(A\right)$, and define $g, h\colon B \to B$ such that $g$ is the identity function, and that $h\left(x\right) = x$ for every $x \in B,$ except that $h\left(b\right)$ 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 maps are homomorphisms: let $C$ be the cokernel of $f$, and $g\colon B \to C$ be the canonical map, such that $g\left(f\left(A\right)\right) = 0$. 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 relatio ...
$\sim$ on $X$ by $a \sim b$ if and only if $f\left(a\right) = f\left(b\right)$. The relation $\sim$ is called the kernel of $f$. It is a congruence relation 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 ...
$X/$ 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
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 s ...
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 $K$"). 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 learni ...
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 space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s and modules, but is different and has received a specific name in other cases, such as normal subgroup for kernels of group homomorphisms and ideals for kernels of ring homomorphisms (in the case of non-commutative rings, the kernels are the two-sided ideals).

Relational structures

In
model theory 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 ...
, 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 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 sym ...
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 $h\left(uv\right) = h\left(u\right) h\left(v\right)$ 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 monoi ...
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 theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatena ...
. 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 $h\left(x\right) \neq \varepsilon$ for all $x \neq \varepsilon$ in $\Sigma_1^*$. A homomorphism $h \colon \Sigma_1^* \to \Sigma_2^*$ on $\Sigma_1^*$ that satisfies $, h\left(a\right), = k$ for all $a \in \Sigma_1$ is called a $k$''-uniform'' homomorphism. p. 287 If $, h\left(a\right), = 1$ 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 ele ...
generated by $\Sigma$. Here the monoid operation is
concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatena ...
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.

*
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 ...
*
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, morphism ...
*
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 positiv ...

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 i ...
. 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