(The following definitions are meant to be standard and are reproduced for completeness of the question.) A **frame** is a partially ordered set in which every finite subset has a greatest lower bound (“meet”, denoted $x_1\wedge\cdots\wedge x_n$, or $\top$ for the empty meet) and every subset has a least upper bound (“join”, denoted $\bigvee\{x_i : i\in I\}$), and such that the latter distributes over the former, which amounts to demanding $z \wedge \bigvee \{x_i : i \in I\} = \bigvee\{(z\wedge x_i) : i \in I\}$ (in fact, arbitrary meets exist in a frame, but they generally fail to distribute over joins). (An important source of examples of frames are the lattice of open sets on a topological space, in which case finite meets and arbitrary joins are simply finite intersections and arbitrary unions.) The **Heyting operation** on a frame $L$, denoted $\Rightarrow$, is defined as $(x\Rightarrow z) := \bigvee\{y : x\wedge y \leq z\}$ (this sup is, in fact, a max). A **nucleus** on a frame $L$ is a map $j\colon L\to L$ such that ① $j(x\wedge y) = j(x)\wedge j(y)$ (from which it follows that $j$ is order-preserving), ② $x \leq j(x)$, and ③ $j(j(x)) = j(x)$ (these three conditions are understood universally quantified over free variables in $L$). The set of nuclei on $L$, with the pointwise order (viꝫ., $j_1\leq j_2$ when $j_1(x)\leq j_2(x)$ for all $x\in L$) is itself a frame, denoted $N(L)$; its top element is $x \mapsto \top_L$ and bottom $x \mapsto x$. Meet of nuclei can be defined pointwise: $(j_1\wedge j_2)(x) = j_1(x) \wedge j_2(x)$ (in fact, the same holds for arbitrary meets). Joins of nuclei, however, are more difficult to compute in general, although it is true that the fixset $L_j := \operatorname{im} j = \{x\in L : j(x)=x\}$ of $j := \bigvee\{j_i : i\in I\}$ is $\bigcap_{i\in I} L_{j_i}$ (so $j(x)$ can be expressed as the smallest element $\geq x$ of $L$ which is fixed by every $j_i$). See Escardó, “Joins in the Frame of Nuclei” (2003) for details and references about nuclei in general and how to compute their join.

My question concerns the computation of the Heyting operation in the frame $N(L)$ of nuclei over a frame $L$. So $j_1\Rightarrow j_2$ is defined as the largest nucleus $k$ such that $j_1(x) \wedge k(x) \leq j_2(x)$ for all $x$ (in particular, we have $(j_1\Rightarrow j_2)(x) \leq (j_1(x) \Rightarrow j_2(x))$). But this is not an equality in general.

To provide at least one interesting example of this, for an arbitrary element $a\in L$ we two naturally occurring nuclei, $j_a\colon x\mapsto a\vee x$ (the “complementary closed sublocale” nucleus associated to $a$) and $j^a\colon x\mapsto (a\Rightarrow x)$. They turn out to be complementary in $N(L)$ (see, e.g., Fourman & Scott, “Sheaves in Logic”, p. 302–401 in Fourman, Mulvey & Scott, *Applications of Sheaves (Durham 1977)*, Springer LNM **753** (1979), §2.18), in the sense that $j_a \vee j^a = \top_{N(L)}$ and $j_a \wedge j^a = \bot_{N(L)}$; in particular, we have $j^a = (j_a \Rightarrow \bot_{N(L)})$, which is unsurprising, but also $j_a = (j^a \Rightarrow \bot_{N(L)})$, which is more surprising as the inequality $a\vee x \leq ((a\Rightarrow x) \Rightarrow x)$ is generally not an equality (the RHS here is not a nucleus in $x$, although it *is* a nucleus in $a$ which caused me some level of confusion).

**Question:** How can we compute the value $(j_1\Rightarrow j_2)(x)$ of the Heyting operation for $j_1,j_2 \in N(L)$ two nuclei (and $x\in L$)? Or how can we, at least, say something beyond the inequality $(j_1\Rightarrow j_2)(x) \leq (j_1(x) \Rightarrow j_2(x))$ remarked above? Pretty much anything that can be said about $(j_1\Rightarrow j_2)(x)$ interests me, including the special cases $j_2 = \bot_{N(L)}$, and/or when $L$ is the lattice of open sets on a topological space $X$ and $j_1,j_2$ are defined by reflecting to open subsets of subspaces $E_1,E_2 \subseteq X$.