On Boolean, Heyting and Brouwerian Algebras

Abstract

Several models for logic have been developed in previous century. Each model has its own algebra of truth values. Frequently used algebras for truth values are Boolean Algebra for classical logic and Heyting Algebra and Brouwerian algebra for intuitionist logic logic and Brazilian logic respectively. Each of these algebras is a distributive lattice. In this paper we consider lattices which admit certain binary operations that force distributivity. For Boolean algebra this binary operation is induced by an endofunction which turns out to be negation for the Boolean algebra. These binary operations and corresponding negations for Heyting algebras and Browerian algebras, are discussed in detail. At the end we give a necessary and sufficient condition for a lattice to be a Boolean algebra.