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
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: