Implications in pseudocomplemented and Stone 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.