HomoMorphism

A homomorphism, from one mathematical object to another of the same kind, is a function 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 OneToOne 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.

Given a homomorphism f:X->Y, we can define an EquivalenceRelation on X by a~b iff f(a)=f(b). The quotient (set of all equivalence classes) X/~ can be defined to have an object-structure in a natural way. We then find that the image of X is necessarily isomorphic to X/~.