Homomorphism
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.
- 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.
- Comorphism[?] is a term sometimes used in category theory.
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