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 $\displaystyle \alpha^{-1} \left({a'}\right)$ coincides with $\displaystyle \overline S$; for, in this case, the mapping $\displaystyle \overline a$ is 1-1. Thus if $\displaystyle \overline a \overline \alpha = \overline b \overline \alpha$, then $\displaystyle a \alpha = b \alpha$ and $\displaystyle a \sim b$. Hence $\displaystyle \overline a = \overline b$. Thus we obtain here a factorization $\displaystyle \alpha = \nu \overline \alpha$ where $\displaystyle \overline \alpha$ is 1-1 onto $\displaystyle T$ and $\displaystyle \nu$ is the natural mapping."


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


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 $\displaystyle \alpha^{-1} \left({a'}\right)$ coincides with $\displaystyle \overline S$".

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

What am I missing here?