A homomorphism, (or sometimes simply morphism) from one mathematical object to another of the same kind, is a mapping that is compatible with all relevant structure. The notion of morphism is studied abstractly in category theory.
For example, if one object consists of a set X with an ordering <= and the other object consists of a set Y with an ordering [=, then it must hold for the function f : X -> Y that
- if u <= v then f(u) [= f(v).
Or, if on these sets the binary operations * and @ are defined, respectively, then it must hold that
- f(u) @ f(v) = f(u * v).
An example of homomorphism is given by group homomorphism.
A homomorphism which is also a bijection is called an isomorphism; two isomorphic objects are completely indistinguishable as far as the structure in question is concerned. A homomorphism from a set to itself is called an endomorphism, and if it is also an isomorphism is called an automorphism.
Any homomorphism f : X -> Y defines an equivalence relation on X by a ~ b iff f(a) = f(b). The quotient set X / ~ can then be given an object-structure in a natural way, e.g. [x] * [y] =[x * y]. In that case the image of X is necessarily isomorphic to X / ~. Note in some cases (e.g. groups) a single equivalence class U suffices to specify the structure of the quotient, so we write it X / U.