(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.)
I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements in the frame $N(L)$ of nuclei on $L$, meaning that $(\neg\neg j) = j$, where $\neg\neg\colon N(L)\to N(L)$ is the double-negation operation on $N(L)$ (this is not just the pointwise operation: see the aforementioned question for clarifications).
(From the answer I received to the aforementioned question, a necessary and sufficient condition for a nucleus to be regular is that it satisfies $j(z) = \bigwedge_{y\geq z}((\bigwedge_{x\geq y} (j(x)\Rightarrow x))\Rightarrow y)$ for all $z\in L$, but this does not appear very workable. Obvious examples of such nuclei are $j_a\colon x \mapsto a\vee x$ and $j^a\colon x \mapsto (a\Rightarrow x)$ for any $a\in L$, as well as meets like $j_a\wedge j^b$, but these nuclei are actually complemented, not just regular.)
The set $N(L)_{\neg\neg}$ of these regular nuclei is a complete Boolean algebra, namely the Booleanization (=largest Boolean quotient) of $N(L)$, and the map $L \to N(L)_{\neg\neg}$ given by $a \mapsto j_a$ thus provides an embedding of any frame $L$ into a complete Boolean algebra which, even though it is arguably not “natural”, still seems worthy of interest. (E.g., it is not obvious, constructively, that every frame embeds in a Boolean algebra, and this provides a proof.)
So anyway, my question is: where might I learn more about such nuclei, about the sublocales they define, or about the Boolean algebra $N(L)_{\neg\neg}$ they form?
I don't even know what to call such sublocales (the word “regular” isn't good because “regular locale” means something different and “regular sublocale” should probably be reserved for sublocales that, when viewed as locales on their own right, are regular), so suggestions are welcome, especially if they are already established terms (so I can use them in a search engine).
The reference I found that came closest to discussing this in any way is the preprint by Picado, Pultr & Tozzi, “Joins of closed sublocales” (Universidade de Coimbra preprint 16–39), which actually discusses a different frame, namely that of sublocales that are joins of closed sublocales, which in a certain case (when $L$ is “subfit”) coincides with the one I'm asking about (op. cit., theorem 3.4). Also, we learn that if $L$ is the frame of open sets of a (sober?) $T_1$ topological space $X$, then the Boolean algebra $N(L)_{\neg\neg}$ I'm asking about can be identified with that $\mathscr{P}(X)$ of all subspaces of $X$ (op. cit., theorem 4.3).
So essentially any other reference giving $N(L)_{\neg\neg}$ more than a passing mention interests me. For example, a characterization more simple than the formula $j(z) = \bigwedge_{y\geq z}((\bigwedge_{x\geq y} (j(x)\Rightarrow x))\Rightarrow y)$ written above would be highly interesting.
Remarkably, in J. Todd Wilson's thesis (“The Assembly Tower…”, 1994), there is a lengthy discussion (chapter 6) of what he calls “regular operators” on a frame, which turn out to be equivalent to regular nuclei on $N(L)$ itself, i.e., elements of the Boolean algebra $N^2(L)_{\neg\neg}$ (where $N^2(L)$ means $N(N(L))$), but I couldn't find anything about $N(L)_{\neg\neg}$, and I wasn't able to find a way to use his results on $N^2(L)_{\neg\neg}$ to say something about $N(L)_{\neg\neg}$.