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.

**[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.