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Math and science::Algebra::Aluffi

Canonical decomposition

Any function f:AB can be decomposed into a [type of function], followed by an [type of function], followed by an [type of function].

The surjection will be a projection to a partition of A induced by f. The isomorphism will be from this set to [what set?]. The injection will be [what function?] back to B.

Equivalence relation induced by a function

Let f:AB be a function. f induces an equivalence relation on A as follows: For all a,aA,

aaf(a)=f(a).

Theorem. Canonical decomposition

Let f:AB be any function, and define as above. Then f decomposes as follows:

First we have the canonical projection AA/. The last function is the inclusion imfB. The bijection in the middle is defined as:

[f~(?)=?]

This theorem states that the above diagram for f's decomposition computes and that f~ is a valid function and is a bijection.

Well-defined

There is ambiguity as to which elements of A are chosen to represent the equivalence classes in A . As a consequence, this theorem should be accompanied by a proof to show that any choice available leads to the same result. Such a proof is said to be a verification of the theorem being well-defined. Aluffi does exactly this, and it is a good example of such proofs.