Two Axioms for Implication Algebras
A. Gareau, Ranganathan Padmanabhan · 2006
It is well-known that the implicational fragment of the classical propositional calculus has a single axiom. By contrast, here we show that the corresponding equational class defined by the implicational reduct of Boolean algebra cannot be defined by a single axiom. However, it can be defined by two identities. By a deep theorem of Alfred Tarski, it follows that this variety has an independent basis with n identities for all n > 1. Furthermore, it follows that no equational theory defined by any of the six well-known orthomodular implications is one-based. The implicational fragment of the 2-element Boolean algebra is the class of all algebras of type 〈2〉 having a single binary operation → with the interpretation that x → y = x′∨y. Abbott [1] first defined these implication algebras by the following three identities: (x → y) → x = x (1) (x → y) → y = (y → x) → x (2) x → (y → z) = y → (x → z) (3) In 1948, Lukasiewicz [4] proved that the implicational fragment of 2-valued logic is one-based, that is, it can be defined by a single axiom. The shortest single axiom is thanks to Tursman [8]: i(i(i(x, y), z), i(i(z, x), i(u, x))). It is natural to ask whether this is also possible for the implicational reduct of Boolean algebras. In Appendix 4 of Gratzer [2], W. Taylor claims D.H. Potts proved the following result in Potts [6], however, Potts’ paper makes no mention of implication algebras and is focused entirely on semilattices. Printed July 23, 2006 2001 Mathematics Subject Classification: Primary, 06C15; Secondary, 68T15