QUANTUM AND CLASSICAL IMPLICATION ALGEBRAS WITH PRIMITIVE IMPLICATIONS

Mladen Pavicic^1,2 and Norman D. Megill³

¹Atominstitute of the Austrian Universities, Schüttelstraße 115, A-1020 Wien, Austria; pavicic@ati.ac.at

² Department of Mathematics, University of Zagreb, GF, Kaciceva 26, POB-217, HR-10001 Zagreb, Croatia; mpavicic@faust.irb.hr; http://m3k.grad.hr/pavicic

³ Locke Lane, Lexington, MA 02173, U. S. A.; nm@alum.mit.edu

Abstract. Join in an orthomodular lattice is obtained in the same form for all five quantum implications. The form holds for the classical implication in a distributive lattice as well. Even more, the definition added to an ortholattice makes it orthomodular for quantum implications and distributive for the classical one. Based on this result a quantum implication algebra with a single primitive - and in this sense unique - implication is formulated. A corresponding classical implication algebra is also formulated. The algebras are shown to be special cases of a universal implication algebra.

PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd

Keywords: implication algebra, orthomodular lattices, quantum logic, classical logic.