Orthomodular lattices as implication algebras

Abstract

Recent progress has been made in the study of implication connectives on orthomodular lattices (see [6] and [7]). Some of this research has been motivated by the delicate question of the appropriate form of the conditional sentence in a quantum logic. The purpose of this paper is to show that an orthomodular lattice can be axiomatized in terms of an "implication connective" on a set with a distinguished element. That such axiomatizations exist for Boolean algebras, a special family of orthomodular lattices, is well known to both logicians and mathematicians (see for example [1]). Finch in [3] began with an orthocomplemented partially ordered set with an implication connective and negation operator satisfying certain axioms, and proved the poset is actually an orthomodular lattice. We extend the work of Finch by beginning with a set P with an element 0 in P and with an abstractly given implication connective D. The triple (P, 0, D) is called a quantum implication algebra and is an algebra in the usual sense. No assumption is made about a partial order on P. Nevertheless, we prove that the theory of quantum implication algebras is co-extensive with the theory of orthomodular lattices. In all fairness, both Finch and I are using the original ideas of D. J. Foulis (see [4] and [5]) to accomplish the above mentioned results. I'm sure both of us are indebted to him for his insights.