I'm studying N. Jacobson's "Lectures in Advanced Algebra" (originally published in 1951, apparently a classic), and have encountered this puzzling paragraph:

:"This type of factorization of mappings ... is particularly useful when the set of inverse images \alpha^{-1} \left({a'}\right) coincides with \overline S; for, in this case, the mapping \overline a is 1-1. Thus if \overline a \overline \alpha = \overline b \overline \alpha, then a \alpha = b \alpha and a \sim b. Hence \overline a = \overline b. Thus we obtain here a factorization \alpha = \nu \overline \alpha where \overline \alpha is 1-1 onto T and \nu is the natural mapping."


Note that in the above, Jacobson uses:
* \alpha for a general mapping from S to T;
* a' for the Image of a representative element a of S under \alpha;
* \overline S for the quotient set defined by the equivalence induced by \alpha;
* \nu for the quotient mapping from S \to \overline S;
* \overline a and \overline b for representative elements of \overline S;
* \overline \alpha for the renaming mapping \overline S \to T.


The above, then, is a fairly terse proof of the Quotient Theorem for Surjections: the fact that a surjection can be factored into the canonical surjection and the renaming mapping, the latter of which is a bijection.


The puzzling statement is the one: "when the set of inverse images \alpha^{-1} \left({a'}\right) coincides with \overline S".

Surely, as \alpha is a mapping, the quotient set of S induced by \alpha always forms a partition of S, and so the "set of inverse images MATH]\alpha^{-1} \left({a'}\right)[/tex] will always coincide with \overline S?

What am I missing here?