en.wikipedia.org/wiki/Currying
1 correction found
Distributive Heyting algebras are Boolean algebras
This is incorrect because Heyting algebras are already distributive by definition, but they are not generally Boolean. A Heyting algebra is Boolean only under extra conditions such as the law of excluded middle.
Full reasoning
This claim is false because distributivity does not make a Heyting algebra Boolean.
A Heyting algebra is the algebraic semantics of intuitionistic propositional logic, while Boolean algebras model classical propositional logic. Authoritative references state that:
- Any Heyting algebra is a distributive lattice.
- A Heyting algebra becomes Boolean only if it satisfies an additional condition such as excluded middle (
x ∨ ¬x = 1) or equivalently double negation elimination (¬¬x = x).
So the article's wording reverses the actual relationship. The correct relationship is closer to:
- Every Boolean algebra is a Heyting algebra; but
- not every Heyting algebra is Boolean.
One standard family of counterexamples comes from topologies/open-set lattices: these form Heyting algebras, but only special cases are Boolean. As nLab notes, topologies that are Boolean algebras are "the exception rather than the rule."
2 sources
- Heyting algebra in nLab
Any Heyting algebra is a distributive lattice. ... A Heyting algebra satisfying ... x ∨ ¬x = 1 ... is precisely a Boolean algebra.
- Pseudo-Boolean algebra - Encyclopedia of Mathematics
Pseudo-Boolean algebras serve as algebraic models of Heyting's intuitionistic propositional calculus ... in the same way that Boolean algebras characterize classical propositional calculus. ... Every pseudo-Boolean algebra is a distributive lattice.