endomorphism

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an endomorphism is a morphism from a
mathematical object A mathematical object is an abstract concept arising in mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantit ...
to itself. An endomorphism that is also an
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
is 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 map (mathematics), mapping the object to itself while preserving all of its structure. The Set (m ...
. For example, an endomorphism of a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is a
linear map In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, and an endomorphism of a group is a
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
. 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), ''Categories'' (Aristotle) *Category (Kant) ...
. In the
category of sets In the mathematical field 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 foundatio ...
, 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 ar ...
, the full transformation monoid, and denoted (or to emphasize the category ).

# Automorphisms

An
invertible In mathematics, the concept of an inverse element generalises the concepts of additive inverse, opposite () and Multiplicative inverse, reciprocal () of numbers. Given an operation (mathematics), operation denoted here , and an identity element ...
endomorphism of is called 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 map (mathematics), mapping the object to itself while preserving all of its structure. The Set (m ...
. The set of all automorphisms is a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
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 (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
, , 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 In mathematics, the endomorphisms of an abelian group ''X'' form a ring (mathematics), ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms ari ...
). 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 of ...
entries. The endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a
preadditive category In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab. That is, an Ab-cate ...
. The endomorphisms of a nonabelian group generate an algebraic structure known as a
near-ring In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
. 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;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
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set,
acting Acting is an activity in which a story is told by means of its Enactment (psychology), enactment by an actor or actress who adopts a Character (arts), character—in theatre, television, film, radio, or any other medium that makes use of the ...
on the elements, and allowing the notion of element
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...
s to be defined, etc. Depending on the additional structure defined for the category at hand (
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
. A homomorphic endofunction is an endomorphism. Let be an arbitrary set. Among endofunctions on one finds
permutation In mathematics, a permutation of a Set (mathematics), 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 ...
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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
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-dimensiona ...
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 n ...
the floor of has its image equal to its codomain and is not invertible. Finite endofunctions are equivalent to directed pseudoforests. 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.