Heyting implication ($a\to b$)

In a Heyting algebra, $a\to b$ is the greatest $x$ with $x\wedge a\le b$. Thus $x\le a\to b$ if and only if $x\wedge a\le b$. Every frame has this operation, obtained by taking the join of all such $x$. The open nucleus attached to $a$ is $x\mapsto a\to x$.

The pseudocomplement of $a$ is the greatest element disjoint from $a$. It need not satisfy $a\vee \neg a=1$. In a frame, double pseudocomplementation defines the Booleanization of a locale; its fixed elements are regular open elements.