In mathematics, an endomorphism is a morphism from a

mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...

to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism 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 ...

is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in 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) ...

. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms 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 ...

, the

full transformation monoid In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transfo ...

, and denoted (or to emphasize the category ).

Automorphisms

An invertible endomorphism of is called an automorphism. The set of all automorphisms is a

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication:

Endomorphism rings

Any two endomorphisms of an

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 com ...

, , can be added together by the rule . Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of is the ring of all matrices with

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 ...

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 near-ring. Every ring with one is the endomorphism ring of its

regular module In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular re ...

, and so is a subring of an endomorphism ring of an abelian group;Jacobson (2009), p. 162, Theorem 3.2. however there are rings that are not the endomorphism ring of any abelian group.

Operator theory

In any concrete category, especially for

s, endomorphisms are maps from a set into itself, and may be interpreted as

unary operator 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 ...

s on that set, acting on the elements, and allowing the notion of element

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...

s to be defined, etc. Depending on the additional structure defined for the category at hand (

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, metric, ...), such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory.

Endofunctions

An endofunction is a function whose domain is equal to its

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 ...

. A homomorphic endofunction is an endomorphism. Let be an arbitrary set. Among endofunctions on one finds

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

s of and constant functions associating to every in the same element in . Every permutation of has the codomain equal to its domain and is bijective and invertible. If has more than one element, a constant function on has an

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...

that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

the floor of has its image equal to its codomain and is not invertible. Finite endofunctions are equivalent to

directed pseudoforest In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every connected component has at most one cycle. Th ...

s. For sets of size there are endofunctions on the set. Particular examples of bijective endofunctions are the involutions; i.e., the functions coinciding with their inverses.

Notes

References

External links

* {{springer, title=Endomorphism, id=p/e035600 Morphisms