Homomorphism

This word must not be confused with homeomorphism.

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 homomorphism is studied abstractly in universal algebra, and that is the viewpoint taken in this article. A more general 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).

Examples of morphisms are given by group homomorphisms, ring homomorphisms, linear operators, continuous maps etc.

Any homomorphism f: X -> Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b). In the general case, this ~ is called the kernel of f. 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 in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems. Note in some cases (e.g. groups or rings), a single equivalence class K suffices to specify the structure of the quotient, so we write it X/K. Also in these cases, it is K, rather than ~, that is called the kernel of f.

Variants and subclasses of homomorphism:

A homomorphism which is also a bijection such that its inverse is also a homomorphism 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.
A homomorphism which is surjective is called an epimorphism.
A homomorphism which is injective is called a monomorphism.

(The above terms are used similarly in category theory as well as in universal algebra, but the definitions in category theory are more subtle; see the article on morphism for those.)

A homeomorphism preserves topological properties, and is really a kind of isomorphism.
A diffeomorphism preserves differential topological properties, and is really a kind of isomorphism.

(The above terms don't really belong to the subject of universal algebra, but they are listed here anyway in case you are looking for them. In particular, note that "homeomorphism" does not mean quite the same thing as "homomorphism".)

Comorphism is a term sometimes used in category theory.