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# Homomorphism

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A homomorphism, (or sometimes simply morphism) from one mathematical object to another of the same kind, is a mapping that preserves all relevant structure.

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

  f(u) [= f(v) iff u <= 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).

Such an homomorphism is called a 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 Y suffices to specify the structure of the quotient, so we write it X / Y.

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Last edited July 31, 2001 3:00 am (diff)
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