# zbMATH — the first resource for mathematics

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.

##### MSC:
 06C15 Complemented lattices, orthocomplemented lattices and posets 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic
Full Text:
##### References:
  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  Bennett, M. K. and Foulis, D. J. (1990) Superposition in quantum and classical mechanics,Foundations of Physics 20(6), 733-744.  Beren, L. (1984)Orthomodular Lattices, D. Reidel, Dordrecht.  Birkhoff, G. and von Neumann, J. (1936) The logic of quantum mechanics,Ann. of Math. 37, 823-843. · JFM 62.1061.04  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  Foulis, D. J. (1989) Coupled physical systems,Foundations of Physics 7, 905-922.  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  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.  Greechie, R. J. (1971) Orthomodular lattices admitting no states,J. Combinatorial Theory 10, 119-132. · Zbl 0219.06007  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.  Halmos, P. R. (1963)Lectures on Boolean Algebras, Van Nostrand, Princeton. · Zbl 0114.01603  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.  Kalmbach, G. (1983)Orthomodular Lattices, Academic Press, N.Y. · Zbl 0512.06011  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  Lock, P. F. (1981) Categories of Manuals, Ph.D. Dissertation, Univ. of Mass., Amherst, MA.  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.  Randall, C. H. and Foulis, D. J. (1979)New Definitions and Results, Univ. of Mass. Mimeographed Notes, Sept. 1979.  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  Sikorski, R. (1960)Boolean Algebras, Springer-Verlag, Berlin/Göttingen/Heidelberg. · Zbl 0087.02503  Stone, M. H. (1936) The theory of representations for a Boolean algebra,Trans. Amer. Math. Soc. 40, 37-111. · Zbl 0014.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.