Nucleus on a frame ($j:A\to A$)

A nucleus is an inflationary, idempotent, finite-meet-preserving map on a frame. Its fixed elements form a frame, whose joins are obtained by applying $j$ to ambient joins. The induced locale is a sublocale. The nuclei on a frame themselves form a frame in pointwise order; their meets are pointwise, but their joins usually are not.

For a frame homomorphism $h:B\to A$, nuclei are equivalent when their restrictions to $h(B)$ agree. The least representative of the class of $j$ isThe least representatives form a frame generated by $c(A)$ and $o(h(B))$. To see minimality, each summand is below $j$ and its value at $hb$ is $j(hb)$; any equivalent nucleus therefore bounds every summand. This construction gives the pushout of $B\to N(B)$ along $h$.

This nucleus on a frame determines the closed sublocale complementary to the open determined by $a$. It is the least nucleus on a frame taking $0$ to $a$. In the frame of nuclei on a frame, $c(a)$ and the open nucleus $o(a)$ are complements.

The Heyting implication defines a nucleus on a frame whose sublocale is the open determined by $a$. It is the least nucleus on a frame sending $a$ to $1$.