Johnson's axioms revisited: Bases for Boolean algebras containing identities of associative type. I
Hanamantagouda P. Sankappanavar
TL;DR
The paper Tackles whether Boolean algebras admit axiom systems that include an associative-type identity beyond the standard associativity. It proves Johnson's third axiom is redundant and constructs explicit $2$-bases and $3$-bases for the variety of Boolean algebras containing each of the $14$ length-$3$ associative-type identities, with several bases convertible to classical propositional logic axiomatizations. It lays out a systematic program to realize a base for every such identity, starting with a $2$-base for $(A6)$, $(A8)$, $(A5)$, and $(A13)$ and a $3$-base for $(A9)$ and $(A1)$, followed by further bases in sequel papers. This work advances the understanding of equational axiomatizations for Boolean algebras by linking associative-type identities to concrete bases and establishing redundancy and independence properties in Johnson-like systems.
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.