Heyting algebra

English

Etymology

After Arend Heyting.

Noun

Heyting algebra (plural Heyting algebras)

1. A bounded lattice equipped with a binary operation ab of implication such that (ab)∧ab, and moreover ab is the greatest such in the sense that if cab then cab.
• 1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page xii:
The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.
If the less-than-or-equal relation in the definition of Heyting algebra is interpreted to mean logical entailment, then (a→b)∧a ≤ b translates to "(a→b), a ⊢ b" which is Modus Ponens. The formulae in the second condition translate to "c, a ⊢ b ⇒ c ⊢ a→b" which is the Deduction Theorem.