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Filters and supports in orthoalgebras. (English) Zbl 0764.03026
Summary: An orthoalgebra, which is a natural generalization of an orthomodular lattice or poset, may be viewed as a “logic” or “proposition system” and, under a well-defined set of circumstances, its elements may be classified according to the Aristotelian modalities: necessary, impossible, possible, and contingent. The necessary propositions band together to form a local filter, that is, a set that intersects every Boolean subalgebra in a filter. In this paper, we give a coherent account of the basic theory of orthoalgebras, define and study filters, local filters, and associated structures, and prove a version of the compactness theorem in classical algebraic logic.

MSC:
03G25 Other algebras related to logic
03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
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[1] Alfsen, E., and Shultz, F. (1976). Non-commutative spectral theory for affine function spaces on convex sets,Memoirs of the American Mathematical Society,172. · Zbl 0337.46013
[2] Beran, L. (1984).Orthomodular Lattices, Reidel/Academia, Dordrecht, Holland. · Zbl 0558.06008
[3] Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823-843. · JFM 62.1061.04
[4] Cook, T. (1985). Banach spaces of weights on quasimanuals,International Journal of Theoretical Physics,24(11), 1113-1131. · Zbl 0579.46006
[5] D’Andrea, A., and De Lucia, P. (1991). The Brooks-Jewett theorem on an Orthomodular lattice,Journal of Mathematical Analysis and Applications,154, 507-522. · Zbl 0727.28008
[6] Foulis, D. (1962). A note on orthomodular lattices,Portugaliae Mathematica,21(1), 65-72. · Zbl 0106.24302
[7] Foulis, D. (1989). Coupled physical systems,Foundations of Physics,7, 905-922.
[8] Foulis, D., and Randall, C. (1981). Operational statistics and tensor products, inInterpretations and Foundations of Quantum Theory, H. Neumann, ed., Vol. 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim, pp. 21-28.
[9] Golfin, A. (1987). Representations and products of lattices, Ph.D. thesis, University of Massachusetts, Amherst, Massachusetts.
[10] Greechie, R., and Gudder, S. (1971). Is a quantum logic a logic?,Helvetica Physica Acta,44, 238-240.
[11] Greechie, R., and Gudder, S. (1973). Quantum logics, inContemporary Research in the Foundations and Philosophy of Quantum Theory, C. Hooker, ed., Reidel, Boston. · Zbl 0279.02015
[12] Gudder, S. (1988).Quantum Probability, Academic Press, Boston. · Zbl 0653.60004
[13] Hardegree, G., and Frazer, P. (1981). Charting the labyrinth of quantum logics, inCurrent Issues in Quantum Logic, E. Beltrametti and B. van Fraassen, eds., Ettore Majorana International Science Series, 8, Plenum Press, New York.
[14] Janowitz, M. (1963). Quantifiers on quasi-orthomodular lattices, Ph.D. thesis, Wayne State University. · Zbl 0144.25303
[15] Jauch, J. (1968).Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts. · Zbl 0166.23301
[16] Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, New York. · Zbl 0512.06011
[17] Kl?y, M., Randall, C., and Foulis, D. (1987). Tensor products and probability weights,International Journal of Theoretical Physics,26(3), 199-219. · Zbl 0641.46049
[18] Lock, P., and Hardegree, G. (1984a). Connections among quantum logics, Part 1, Quantum prepositional logics,International Journal of Theoretical Physics,24(1), 43-53. · Zbl 0592.03051
[19] Lock, P., and Hardegree, G. (1984b). Connections among quantum logics, Part 2, Quantum event logics,International Journal of Theoretical Physics,24(1), 55-61. · Zbl 0592.03052
[20] Piron, C. (1964). Axiomatique quantique,Helvetica Physica Acta,37, 439-468. · Zbl 0141.23204
[21] Piron, C. (1976).Foundations of Quantum Physics, A. Wightman, ed., Benjamin, Reading, Massachusetts. · Zbl 0333.46050
[22] Ramsay, A. (1966). A theorem on two commuting observables,Journal of Mathematics and Mechancis,15(2), 227-234. · Zbl 0143.23003
[23] Randall, C., and Foulis, D. (1979). New definitions and theorems, University of Massachusetts Mimeographed Notes, Amherst, Massachusetts, Autumn 1979.
[24] Randall, C., and Foulis, D. (1981a). Empirical logic and tensor products, inInterpretations and Foundations of Quantum Theory, H. Neumann, ed., Vol. 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim, pp. 9-20. · Zbl 0495.03041
[25] Randall, C., and Foulis, D. (1981b). What are quantum logics and what ought they to be?, inCurrent Issues in Quantum Logic, E. Beltrametti and B. van Fraassen, eds., Ettore Majorana International Science Series 8, Plenum Press, New York, pp. 35-52.
[26] R?ttimann, G. (1979). Non-commutative measure theory, Habilitationsschrift, University of Bern, Switzerland.
[27] R?ttimann, G. (1989). The approximate Jordan-Hahn decomposition,Canadian Journal of Mathematics,41(6), 1124-1146. · Zbl 0699.28001
[28] Sasaki, U. (1954). On orthocomplemented lattices satisfying the exchange axiom,Hiroshima Japan University Journal of Science Series A,17(3), 293-302. · Zbl 0055.25902
[29] Schindler, C. (1986). Decompositions of measures on orthologics, Ph.D. thesis, University of Berne, Switzerland.
[30] Svetlichny, G. (1986). Quantum supports and modal logic,Foundations of Physics,16(12), 1285-1295.
[31] Svetlichny, G. (1990). On the inverse FPR problem: Quantum is classical,Foundations of Physics,20(6), 635-650.
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