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Tensor products of orthoalgebras. (English) Zbl 0798.06015
The following definition of a tensor product of orthoalgebras is proposed: Let \(P\) and \(Q\) be orthoalgebras. We say that a pair \((T,\tau)\) consisting of an orthoalgebra \(T\) and a bimorphism \(\tau: P\times Q\to T\) is a tensor product of \(P\) and \(Q\) iff the following conditions are satisfied: (i) If \(L\) is an orthoalgebra and \(B: P\times Q\to L\) is a bimorphism, there exists a morphism \(\varphi: T\to L\) such that \(B= \varphi\circ\tau\). (ii) Every element of \(T\) is a finite orthogonal sum of elements of the form \(\tau(p,q)\) with \(p\in P\) and \(q\in Q\).
It is shown that the tensor product need not exist. A sufficient condition for its existence are sufficiently large spaces of probability measures: for instance, two unital orthoalgebras have always a tensor product.

06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
Full Text: DOI
[1] Beltrametti, E. and Cassinelli, G. (1981) The Logic of Quantum Mechanics, in Gian-Carlo Rota (ed.),Encyclopedia of Mathematics and its Applications 15, Addison-Wesley, Reading, MA. · Zbl 0491.03023
[2] Bennett, M. K. and Foulis, D. J. (1990) Superposition in quantum and classical mechanics,Foundations of Physics 20(6), 733-744.
[3] Beren, L. (1984)Orthomodular Lattices, D. Reidel, Dordrecht.
[4] Birkhoff, G. and von Neumann, J. (1936) The logic of quantum mechanics,Ann. of Math. 37, 823-843. · JFM 62.1061.04
[5] Foulis, D. J. and Randall, C. H. (1981) Empirical logic and tensor products, in H. Neumann (ed.),Interpretations and Foundations of Quantum Theory, Wissenschaftsverlag, Bibliographisches Institut 5, Mannheim/Wien/Zürich, pp. 9-20. · Zbl 0495.03041
[6] Foulis, D. J. (1989) Coupled physical systems,Foundations of Physics 7, 905-922.
[7] Foulis, D. J., Greechie, R. J. and Rüttimann, G. T. (1992) Filters and supports in orthoalgebras,Int. J. of Theoretical Physics 31(5), 789-807. · Zbl 0764.03026
[8] Foulis, D. J. (1992) Tensor products of quantum logics, to appear in the Proceedings of SymposiumThe Interpretation of Quantum Theory: Where Do We Stand?, April, 1992, sponsored by the Instituto Della Enciclopedia Italiana.
[9] Greechie, R. J. (1971) Orthomodular lattices admitting no states,J. Combinatorial Theory 10, 119-132. · Zbl 0219.06007
[10] Greechie, R. J. and Gudder, S. P. (1975) Quantum logics, in C. A. Hooker (ed.),The Logico-Algebraic Approach to Quantum Mechanics, Volume I:Historical Evolution, D. Reidel, Dordrecht.
[11] Halmos, P. R. (1963)Lectures on Boolean Algebras, Van Nostrand, Princeton. · Zbl 0114.01603
[12] Hardegree, G. and Frazer, P. (1981) Charting the labyrinth of quantum logics, in E. Beltrametti and B. van Fraassen (eds),Current Issues in Quantum Logic, Ettore Majorana International Science Series,8, Plenum Press, N.Y.
[13] Kalmbach, G. (1983)Orthomodular Lattices, Academic Press, N.Y. · Zbl 0512.06011
[14] Kläy, M. P., Randall, C. H. and Foulis, D. J. (1987) Tensor products and probability weights,Int. J. of Theoretical Physics 26(3), 199-219. · Zbl 0641.46049
[15] Lock, P. F. (1981) Categories of Manuals, Ph.D. Dissertation, Univ. of Mass., Amherst, MA.
[16] Lock, R. H. (1990) The tensor product of generalized sample spaces which admit a unital set of dispersion-free weights,Foundations of Physics 20(5), 477-498.
[17] Randall, C. H. and Foulis, D. J. (1979)New Definitions and Results, Univ. of Mass. Mimeographed Notes, Sept. 1979.
[18] Randall, C. H. and Foulis, D. J. (1981) Operational statistics and tensor products, in H. Neumann (ed.),Interpretations and Foundations of Quantum Theory, Band 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim/Wien/Zürich, pp. 21-28. · Zbl 0495.03042
[19] Sikorski, R. (1960)Boolean Algebras, Springer-Verlag, Berlin/Göttingen/Heidelberg. · Zbl 0087.02503
[20] Stone, M. H. (1936) The theory of representations for a Boolean algebra,Trans. Amer. Math. Soc. 40, 37-111. · Zbl 0014.34002
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