Effect algebra counterexamples. (English) Zbl 0890.03035

Gudder, Stanley; Greechie, Richard

Math. Slovaca 46, No. 4, 317-325 (1996).

Two effect algebra counterexamples are presented. The first shows that the standard effect algebra of operators on a Hilbert space is not a lattice and the second shows that the tensor product of two effect algebras need not exist.

MSC:

03G12	Quantum logic
81P10	Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)

Keywords:

effect algebra; D-poset; lattice; tensor product

References:

[1]	AERTS D., DAUBECHIES I.: Physical justification of using tensor products to describe physical systems as one joint system. Helv. Phys. Acta 51 (1978), 661 675.
[2]	CATTANEO G., NISTICO G.: Brouwer-Zadeh posets and three-valued Lukasiewicz posets. Internat. J. Fuzzy. Sets Sys. 33 (1989), 165-190. · Zbl 0682.03036 · doi:10.1016/0165-0114(89)90239-X
[3]	DAVIES E. B.: Quantum Theory of Open Systems. Academic Press, London, 1976. · Zbl 0388.46044
[4]	DAVIES E. B.-LEWIS J. T.: An operational approach to quantum probability. Comm. Math. Phys. 17 (1970), 239-260. · Zbl 0194.58304 · doi:10.1007/BF01647093
[5]	DVUREČENSKIJ A.: Tensor product of difference posets. Trans. Amer. Math. Soc. 347 (1995), 1043-1057. · Zbl 0859.03031 · doi:10.2307/2154888
[6]	DVUREČENSKIJ A., PULMANNOVÁ S.: Tensor products of D-posets and D-test spaces. Rep. Math. Phys. 34 (1994), 4251-4275. · Zbl 0830.03031 · doi:10.1016/0034-4877(94)90001-9
[7]	FOULIS D.: Coupled physical systems. Found. Phys. 19 (1989), 905-922.
[8]	FOULIS D., BENNETT M. K.: Tensor products of orthoalgebras. Order 10 (1993). 271-282. · Zbl 0798.06015 · doi:10.1007/BF01110548
[9]	FOULIS D., BENNETT M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325-1346. · Zbl 1213.06004 · doi:10.1007/BF02283036
[10]	GIUNTINI R. GREULING H.: Toward a formal language for unsharp properties. Found. Phys. 19 (1989), 931-945.
[11]	GREECHIE R.: Orthomodular lattices admitting no states. J. Combin. Theory 10 (1971). 119-132. · Zbl 0219.06007 · doi:10.1016/0097-3165(71)90015-X
[12]	HOLEVO A. S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland. Amsterdam, 1982. · Zbl 0497.46053
[13]	KADISON R.: Order properties of bounded, self-adjoint operators. Proc. Amer. Math. Soc. 34 (1951), 505-510. · Zbl 0043.11501 · doi:10.2307/2031784
[14]	KAKUTANI S.: Concrete representatum of abstract (M)-spaces. Ann. of Math. 42 ( 1941). 994 1024.
[15]	KLÄY M., RANDALL C., FOULIS D.: Tensor products and probability weights. Internat. J. Theoret. Phys. 26 (1987), 199-216. · Zbl 0641.46049 · doi:10.1007/BF00668911
[16]	KRAUS K.: States, Effects, and Operations. Springer-Verlag, Berlin. 1983. · Zbl 0545.46049
[17]	KOPKA F., CHOVANEC F.: D-posets. Math. Slovaca 44 (1994), 21-34. · Zbl 0789.03048
[18]	LAHTI P. J.-MACZYNSKI M. J.: Partial order of quantum effects. J. Math. Phys. 36 (1995), 1673-1680. · Zbl 0829.46060 · doi:10.1063/1.531079
[19]	LOCK R.: The tensor product of generalized sample spaces which admit a unital set of dispersion-free weights. Found. Phys. 20 (1990), 477-498.
[20]	LUDWIG G.: Foundations of Quantum Mechanics I. Springer-Verlag, Berlin, 1983. · Zbl 0509.46057
[21]	PULMANNOVA S.: Tensor products of quantum logics. J. Math. Phys. 26 (1985), 1-5.
[22]	WILCE A.: Tensor products of frame manuals. Internat. J. Theoret. Phys. 29 (1990), 805-814. · Zbl 0715.46047 · doi:10.1007/BF00675098