International Journal of Theoretical Physics, Vol. 39, No. 3, 2000 On Some New Operations on Orthomodular Lattices ² Bart D’Hooghe 1 and Jaroslaw Pykacz 2 Received December 8, 1999 Kotas conditionals are used to define six pairs of disjunction- and conjunction- like operations on orthomodular lattices. Although five of them necessarily differ from the lattice operations on elements that are not compatible, they coincide with the lattice operations on all compatible elements of the lattice and they define on the underlying set a partial order relation that coincides with the original one. Some of the new operations are noncommutative on noncompatible elements, but this does not exclude the possibility to endow them with a physical interpretation. The new operations are in general nonassociative, but for some of them a Foulis–Holland-type theorem concerning associativity instead of distributivity holds. The obtained results suggest that these new operations can serve as alternative algebraic models for the logical operations of disjunction and conjunction. 1. INTRODUCTION Garrett Birkhoff and John von Neumann, the founding fathers of quantum logic theory, were not very satisfied with their own proposal of unrestricted treating of lattice operations (meet and join) as algebraic models of logical operations of conjunction and disjunction of experimentally testable proposi- tions about quantum objects. They were aware of the problems that are bound to emerge when considered propositions are not compatible and they wrote in their historic 1936 paper (Birkhoff and von Neumann, 1936): It is worth remarking that in classical mechanics, one can easily define the meet or join of any two experimental propositions as an experimental proposition— ² This paper is dedicated to the memory of Prof. Gottfried T. Ru ¨ttimann. 1 Departement wiskunde, Vrije Universiteit Brussel, 1050 Brussel, Belgium; e-mail: bdhooghe@vub.ac.be. 2 Instytut Matematyki, Uniwersytet Gdan ´ski, 80-952, Gdan ´sk, Poland; e-mail: pykacz@ksinet.univ.gda.pl. 641 0020-7748/00/0300-0641$18.00/0  2000 Plenum Publishing Corporation