Johnson's axioms revisited: Bases for Boolean algebras containing identities of associative type. I

TL;DR

This paper explores alternative axiomatizations of Boolean algebras that include identities of associative type beyond the standard associative law, providing new bases and demonstrating redundancy in Johnson's axioms.

Contribution

It establishes the existence of multiple bases for Boolean algebras containing various associative identities, and shows Johnson's third axiom is redundant.

Findings

Multiple equational bases for Boolean algebras with associative identities

Johnson's third axiom is redundant in the axiomatization

Bases can be adapted for classical propositional logic

Abstract

This paper is inspired by 1892 paper of Johnson, where he has given an axiomatization for the variety of Boolean algebras (equivalently, for classical propositional calculus). The fact that the axioms of Johnson include the associative law, the most well-known identity of associative type of length 3, led us naturally to the question as to whether there are axiom systems for Boolean algebras that include an identity of associative type, other than the associative law. It turns out that the answer to this question is positive in the strongest sense possible. In fact, corresponding to each of the 14 nonequivalent identities of associative type, there is a base containing that identity. Our goal, in this sequence of papers, of which the present paper is the first, is to describe bases for Boolean algebras, each of which contains at least one identity of associative type of length 3. In this paper, we prove, firstly, that the third axiom of Johnson is redundant. Secondly, we give several equational bases, some 2-bases and some 3-bases, for the variety of Boolean algebras, each containing an identity of associative type, other than the associative law. Each of these bases can be easily converted into an axiomatization for the classical propositional logic, as well.