Stone-Cech compactification of locales

Question

I'm reading 'Stone-Cech compactification of locales I' by Banaschewski and Mulvey. I have a question about lemma 6: I don't understand why it suffices to show that the ideals $h(x)$ and $g(x)$ are regular for every $x$. If $L$ were compact then $\bigvee h(x) = \bigvee g(x)$ would imply that h(x) = g(x). But $L$ is not compact.

Answer 1

I think you've found a gap in the paper - which surprises me, I could be missing something. To get the uniqueness you can observe that the image is dense, see IV Theorem 2.2 of Johnstone's "Stone Spaces" for an alternative proof.