Not sure how to fix an error in the Handbook of Categorical Algebra (vol 3)

Question

In the proof of Theorem 1.5.7 (in which it is shown the the nuclei on a locale, when ordered by pointwise partial order, themselves form a locale) in the computation at the bottom of p.35, there is used an inequality

$$ j\Bigl( \bigl( j(b)\Rightarrow k(b)\bigr)\wedge b\Bigr) \Rightarrow k(b) ~~\leq~~ j(b)\Rightarrow k(b)$$

however since $j(bla\wedge b) \leq j(b)$ and $- \Rightarrow k(b)$ is contravariant, I don't think that that's true.

I'm not sure what to do to avoid using this.

Answer 1

You have a point that the last step is unclear. It might have been clearer if the last inequality were written as an equality instead, since that would essentially force the reader to verify that $b = (j(b) \Rightarrow k(b)) \wedge b$, or simply that $b \leq j(b) \Rightarrow k(b)$, which is true since it is equivalent to $b \wedge j(b) \leq k(b)$, which follows from $b \leq k(b)$.