Morphisms

In mathematics, an isomorphism is a structure-preserving mapping or morphism 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, and this is often denoted as . The word is derived .

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: () 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 (1849–1925).

function between groups that preserves multiplication structure

thumb|right|400px|An w:Automorphism|automorphism of the Klein four-group shown as a mapping between two Cayley graphs, a permutation in cycle notation, and a mapping between two Cayley tables.

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not be maps, but they can be composed in a way that is similar to function composition.

In abstract algebra, an endomorphism is a homomorphism from a mathematical object to itself. More generally in category theory, an endomorphism is a morphism from an object in some category to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. frame|right|Orthogonal projection onto a line, , is a [[linear operator on the plane. This is an example of an endomorphism that is not an automorphism.]]

bijection between the vertex set of two graphs

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bijective group homomorphism

mapping between two rings which respects the structure

a structure-preserving correspondence between node-link graphs

bi-universal property in category theory

bijective order-preserving mapping between partially ordered sets

computational problem of determining whether two finite graphs are isomorphic

Z-module homomorphism

scheme morphism such that, with respect to a suitable open cover, is locally of the form Spec(A)→Spec(B) where A is a finitely generated module over B

In functional programming, the concept of catamorphism (from the Ancient Greek: "downwards" and "form, shape") denotes the unique homomorphism from an initial algebra into some other algebra.

type of morphism

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an invertible antihomomorphism, i.e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism.