Tensor products of \(D\)-posets and \(D\)-test spaces. (English) Zbl 0830.03031

Dvurečenskij, Anatolij; Pulmannová, Sylvia

Rep. Math. Phys. 34, No. 3, 251-275 (1994).

A \(D\)-poset (or a difference poset) is a poset \(L\) with the greatest element 1 and a partial binary operation \(\ominus: L\times L\to L\) such that \(b\ominus a\) is defined iff \(a\leq b\) and, for \(a,b, c\in L\), we have (i) \(b\ominus a\leq b\); (ii) \((b\ominus (b\ominus a))=a\); (iii) if \(a\leq b\leq c\), then \(c\ominus b\leq c\ominus a\) and \((c\ominus a) \ominus (c\ominus b)= b\ominus a\). For any \(D\)-poset, there exists a \(D\)- test space [the authors, Rep. Math. Phys. 34, 151-170 (1994; Zbl 0820.03038)]. Using this representation, a tensor product and a state tensor product of \(D\)-posets is introduced and studied, and some examples are presented.

Reviewer: A.Dvurečenskij (Bratislava)

MSC:

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

Keywords:

effect operator; Hilbert space; \(D\)-poset; difference poset; \(D\)-test space; tensor product

Citations:

Zbl 0820.03038

References:

[1]	Busch, P.; Lahti, P. J.; Mittelstaedt, P., The quantum theory of measurement, (Lecture Notes in Physics (1991), Springer: Springer Berlin, Heidelberg, New York, London, Budapest) · Zbl 0868.46051
[2]	Dirac, P. A.M., The Principles of Quantum Mechanics (1930), Clarendon Press: Clarendon Press Oxford · Zbl 0030.04801
[3]	Dvurečenskij, A., Gleason’s Theorem and its Applications (1993), Kluwer Academic Publ: Kluwer Academic Publ Dordrecht, Boston, London · Zbl 0795.46045
[4]	Dvurečenskij, A., Trans. Amer. Math. Soc. (1994), (to appear)
[5]	Dvurečenskij, A.; Pulmannová, S., Inter. J. Theor. Phys., 33, 819 (1994) · Zbl 0806.03040
[6]	Dvurečenskij, A.; Pulmannová, S., Rep. Math. Phys., 34, 151 (1994) · Zbl 0820.03038
[7]	A. Dvurečenskij and S. Pulmannová: Lett. Math. Phys.; A. Dvurečenskij and S. Pulmannová: Lett. Math. Phys.
[8]	Dvurečenskij, A.; Riečan, B., Inter. J. Theor. Phys., 33, 1403 (1994)
[9]	Foulis, D. J.; Bennett, M. K., Order, 19, 271 (1993)
[10]	Foulis, D. J.; Bennett, M. K., Found. Phys. (1994), (to appear)
[11]	Foulis, D. J.; Greechie, R. J.; Rüttimann, G. T., Inter. J. Theor. Phys., 31, 787 (1992)
[12]	Foulis, D. J.; Randall, C. H., J. Math. Phys., 13, 1667 (1972) · Zbl 0287.60002
[13]	Hedlíková, J.; Pulmannová, S., Generalized difference posets and orthoalgebras, Preprint MÚ SAV (1994) · Zbl 0922.06002
[14]	Kalmbach, G.; Riečanová, Z., An axiomatization for abelian relative inverses, Preprint STU Bratislava (1993) · Zbl 0826.08002
[15]	Kolmogorov, A. N., Grundbegriffe der Wahrscheinlichkeitsrechnung (1993), Berlin · Zbl 0278.60002
[16]	Kôpka, F.; Chovanec, F., Math. Slovaca, 44, 21 (1994) · Zbl 0789.03048
[17]	Mesiar, R., Fuzzy difference posets and MV-algebras, Preprint STU (1993), Bratislava
[18]	M. Navara and P. Pták: Difference posets and orthoalgebras (submitted).; M. Navara and P. Pták: Difference posets and orthoalgebras (submitted).
[19]	von Neumann, J., Mathematische Grundlagen der Quantenmechanik (1932), Springer: Springer Berlin · JFM 58.0929.06
[20]	Pulmannová, S., Demonstratio Math., 27 (1994)
[21]	Randall, C. H.; Foulis, D. J., J. Math. Phys., 14, 1472 (1973) · Zbl 0287.60003
[22]	Riečanová, Z.; Bršel, D., Inter. J. Theor. Phys., 33, 133 (1994) · Zbl 0795.03092
[23]	Varadarajan, V. S., Geometry of Quantum Theory, Vol. 1 (1968), van Nostrand: van Nostrand Princeton · Zbl 0155.56802