My stab at Puzzle 22 starting from the definition that \$$\\lnot X = X \\to \\bot \$$ in a
**[Heyting Algebra](https://en.wikipedia.org/wiki/Heyting_algebra)**. This requires that we have a bounded lattice with a smallest and greatest element.

This definition \$$\\lnot X = X \\to \\bot \$$ means that \$$X \\land \\lnot X = \\bot \$$. You then need De Morgan's laws or Law of the Excluded Middle to get \$$X \\lor \\lnot X \\to \\top \$$. In a general Heyting Algebra, these laws aren't true, so I'm not sure if this is allowed to be part of the definition. Turning this into the notation above: $$X \wedge \lnot X = 0$$, where \$$0 \$$ is the smallest element of the bounded lattice.

An example where there is a not operation is on the poset created by the powerset of a finite set, where \$$\\lnot X \$$ is the complement of X.

An example where there isn't a not operation would be the subset \$$\\mathbb{Z}\$$ in \$$\\mathbb{R}\$$ with the usual ordering, as there is no smallest element.