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 instance, if we are concerned about some operation *, homomorphisms must satisfy f(x*y)=f(x)*f(y).

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 EquivalenceRelation on X by a~b iff f(a)=f(b). The quotient 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. GrouP*s*) a single equivalence class Y suffices to specify the structure of the quotient, so we write it X/Y.