Endomorphism
Encyclopedia
In mathematics
, an endomorphism is a morphism
(or homomorphism
) from a mathematical object
to itself. For example, an endomorphism of a vector space
V is a linear map ƒ: V → V, and an endomorphism of a group
G is a group homomorphism
ƒ: G → G. In general, we can talk about endomorphisms in any category
. In the category of sets, endomorphisms are simply functions from a set S into itself.
In any category, the composition
of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid
, denoted End(X) (or EndC(X) to emphasize the category C).
An invertible
endomorphism of X is called an automorphism
. The set of all automorphisms is a subset
of End(X) with a group
structure, called the automorphism group of X and denoted Aut(X). In the following diagram, the arrows denote implication:
Any two endomorphisms of an abelian group
A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under this addition, the endomorphisms of an abelian group form a ring
(the endomorphism ring
). For example, the set of endomorphisms of Zn is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space or module
also form a ring, as do the endomorphisms of any object in a preadditive category
. The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring
. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group, however there are rings which are not the endomorphism ring of any abelian group.
, especially for vector space
s, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits of elements, etc.
Depending on the additional structure defined for the category at hand (topology
, metric
, ...), such operators can have properties like continuity, boundedness
, and so on. More details should be found in the article about operator theory
.
, an endofunction is a function
whose codomain
is equal to its domain
. A homomorphic
endofunction is an endomorphism.
Let S be an arbitrary set. Among endofunctions on S one finds permutation
s of S and constant functions associating to each
a given 
.
Every permutation of S has the codomain equal to its domain and is bijective
and invertible. A constant function on S, if S has more than 1 element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer n the floor of n/2 has its codomain equal to its domain and is not invertible.
Finite endofunctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to 1, and can be easily described.
Particular bijective endofunctions are the involutions, i.e. the functions coinciding with their inverses.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an endomorphism is a morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
(or homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
) from a mathematical object
Mathematical object
In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
to itself. For example, an endomorphism of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V is a linear map ƒ: V → V, and an endomorphism of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
ƒ: G → G. In general, we can talk about endomorphisms in any category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. In the category of sets, endomorphisms are simply functions from a set S into itself.
In any category, the composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
, denoted End(X) (or EndC(X) to emphasize the category C).
An invertible
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
endomorphism of X is called an automorphism
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 of an object forms a group, called the automorphism...
. The set of all automorphisms is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of End(X) with a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
structure, called the automorphism group of X and denoted Aut(X). In the following diagram, the arrows denote implication:
| automorphism 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 of an object forms a group, called the automorphism... |
 |
isomorphism Isomorphism In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are... |
 |
 |
|
| endomorphism |  |
(homo)morphism Homomorphism In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under... |
Any two endomorphisms of an abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under this addition, the endomorphisms of an abelian group form a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
(the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
). For example, the set of endomorphisms of Zn is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space or module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
also form a ring, as do the endomorphisms of any object in a preadditive category
Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
. The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring
Nearring
In mathematics, a near-ring is an algebraic structure similar to a ring, but that satisfies fewer axioms. Near-rings arise naturally from functions on groups.- Definition :...
. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group, however there are rings which are not the endomorphism ring of any abelian group.
Operator theory
In any concrete categoryConcrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
, especially for vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits of elements, etc.
Depending on the additional structure defined for the category at hand (topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
, ...), such operators can have properties like continuity, boundedness
Boundedness
Boundedness or bounded may refer to:*Bounded set, a set that is finite in some sense*Bounded function, a function or sequence whose possible values form a bounded set...
, and so on. More details should be found in the article about operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
.
Endofunctions in mathematics
In mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an endofunction is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
whose codomain
Codomain
In mathematics, the codomain or target set 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...
is equal to its domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
. A homomorphic
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
endofunction is an endomorphism.
Let S be an arbitrary set. Among endofunctions on S one finds permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s of S and constant functions associating to each
a given .Every permutation of S has the codomain equal to its domain and is bijective
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
and invertible. A constant function on S, if S has more than 1 element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer n the floor of n/2 has its codomain equal to its domain and is not invertible.
Finite endofunctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to 1, and can be easily described.
Particular bijective endofunctions are the involutions, i.e. the functions coinciding with their inverses.