Abstract
Kotas conditionals are used to define six pairs of disjunction- andconjunction-like operations on orthomodular lattices. Although five of them necessarily differfrom the lattice operations on elements that are not compatible, they coincidewith the lattice operations on all compatible elements of the lattice and theydefine on the underlying set a partial order relation that coincides with the originalone. Some of the new operations are noncommutative on noncompatible elements,but this does not exclude the possibility to endow them with a physicalinterpretation. The new operations are in general nonassociative, but for someof them a Foulis—Holland-type theorem concerning associativity instead ofdistributivity holds. The obtained results suggest that these new operations canserve as alternative algebraic models for the logical operations of disjunctionand conjunction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Beltrametti, E. G., and Cassinelli, G. (1981). The Logic of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.
Beran, L. (1985). Orthomodular Lattices. Algebraic Approach, Kluwer, Dordrecht.
Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics, Annals of Mathematics, 37, 822-843.
D'Hooghe, B. (n.d.). New operations on orthomodular lattices II. “Disjunctions” and “conjunctions” generated by Kotas conditionals, Annals of Pure and Applied Logic, submitted.
Dorfer, G., Dvure?enskij, A., and Länger, H. (1996). Symmetric difference in orthomodular lattices, Mathematica Slovaca, 46, 435-444.
Foulis, D. J. (1962). A note on orthomodular lattices, Portugaliae Mathematica, 21, 65-72.
Hardegree, G. M. (1974). The conditional in quantum logic, Synthese, 29, 63-80.
Hardegree, G. M. (1979). The conditional in abstract and concrete quantum logic, in The Logico-Algebraic Approach to Quantum Mechanics, II, C. A. Hooker, ed., Reidel, Dordrecht, pp. 49-108.
Hardegree, G. M. (1981). Material implication in orthomodular (and Boolean) lattices, Notre Dame Journal of Formal Logic, 22(2), 163-183.
Holland, S. S., Jr. (1963). A Radon-Nikodym theorem in dimension lattices, Transactions of the American Mathematical Society, 108, 66-87.
Kalmbach, G. (1983). Orthomodular Lattices, Academic Press, London.
Kotas, J. (1967). An axiom system for the modular logic, Studia Logica, 21, 17-38.
Länger, H. (1998). Generalization of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices. Tatra Mountains Mathematical Publications, 15, 97-105.
Mackey, G. W. (1963). The Mathematical Foundations of Quantum Mechanics, Benjamin, New York.
Pykacz, J. (n.d.). New operations on orthomodular lattices I. “Disjunction” and “conjunction” induced by Mackey decompositions, Annals of Pure and Applied Logic, submitted.
Roman, L., and Rumbos, B. (1991). Quantum logic revisited, Foundations of Physics, 21, 727-734.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
D'Hooghe, B., Pykacz, J. On Some New Operations on Orthomodular Lattices. International Journal of Theoretical Physics 39, 641–652 (2000). https://doi.org/10.1023/A:1003637804632
Issue Date:
DOI: https://doi.org/10.1023/A:1003637804632