Implications in pseudocomplemented and Stone lattices
Ivan Chajda, Helmut Länger
TL;DR
The paper investigates how to introduce implication in bounded pseudocomplemented lattices to mimic intuitionistic logic without full Heyting structure. It defines two term-implications, $x\rightarrow y = x^* \vee y$ and $x\Rightarrow y = x^* \vee y^{**}$, and analyzes their algebraic properties, including Modus Ponens-like behavior, contraposition, and how they interact with Stone identities. In Stone lattices, the first implications exhibit adjointness with meet and exchange-like properties, while the second implication satisfies contraposition and quasi-commutativity, though Modus Ponens is not guaranteed in the same manner. The paper also introduces two deductive systems (based on each implication), investigates their closure properties, and defines a relation $\Theta(A)$ that can yield a congruence under Stone identities, linking deductive systems to lattice-theoretic structure. Overall, the work provides an algebraic framework for implications in pseudocomplemented and Stone lattices, offering a bridge between Heyting-like semantics and non-distributive cases, with potential for further axiomatization and applications to logical calculi on such lattices.
Abstract
Two kinds of the connective implication are introduced as term operations of a pseudocomplemented lattice. It is shown that they share a lot of properties with the intuitionistic implication based on Heyting algebras. In particular, if the pseudocomplemented lattice in question is a Stone lattice then the considered implications satisfy some kind of quasi-commutativity, of the exchange property, some version of adjointness with the meet-operation and some kind of the derivation rule Modus Ponens and of the contraposition law. Two kinds of deductive systems are defined and their elementary properties are shown. All investigated concepts are illuminated by examples.