 Solution

The Booleanization of a locale is the sublocale with double-negation nucleus. A flat sublocale has a nucleus preserving finite joins, equivalently its fixed-element set is closed under ambient finite joins. For completeness, if a nucleus preserves finite joins then its fixed points are closed under them. Conversely, if fixed points are closed under joins, $j(a)\vee j(b)$ is fixed and bounds $a\vee b$, giving $j(a\vee b)\le j(a)\vee j(b)$; monotonicity gives equality, and preservation of $0$ is precisely $j(0)=0$. Since $\neg \neg 0=0$, condition (iv) is exactly (iii). Together the preceding implications prove all four characterizations of an extremally disconnected locale.