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27-07-2017 | Foundations

Sum of observables on MV-effect algebras

Author: Anatolij Dvurečenskij

Published in: Soft Computing | Issue 8/2018

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Abstract

Using a one-to-one correspondence between observables and their spectral resolutions, we introduce the sum of any two bounded observables of a \(\sigma \)-MV-effect algebra. This sum is commutative, associative with neutral element. Under the Olson order of observables, the set of bounded observables is a partially ordered semigroup, and the set of sharp observables is even a Dedekind \(\sigma \)-complete \(\ell \)-group with strong unit.

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Bauer H (1996) Probability theory. de Gruyter, BerlinCrossRefMATH

Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–834MathSciNetCrossRefMATH

Butnariu D, Klement EP (1993) Triangular norm-based measures and games with fuzzy coalitions. Kluwer, DordrechtCrossRefMATH

Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, DordrechtCrossRefMATH

Dvurečenskij A (2003) Central elements and Cantor–Bernstein’s theorem for pseudo-effect algebras. J Aust Math Soc 74:121–143MathSciNetCrossRefMATH

Dvurečenskij A (2014) Representable effect algebras and observables. Int J Theor Phys 53:2855–2866. doi:10.1007/s10773-014-2083-z MathSciNetCrossRefMATH

Dvurečenskij A (2016a) Olson order of quantum observables. Int J Theor Phys 55:4896–4912. doi:10.1007/s10773-016-3113 CrossRefMATH

Dvurečenskij A (2016b) Quantum observables and effect algebras. http://arxiv.org/abs/1609.06691

Dvurečenskij A, Kuková M (2014) Observables on quantum structures. Inf Sci 262:215–222. doi:10.1016/j.ins.2013.09.014 MathSciNetCrossRefMATH

Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer, Dordrecht, Ister Science, Bratislava, 541 + xvi pp

Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346MathSciNetCrossRefMATH

Fuchs L (1963) Partially ordered algebraic systems. Pergamon Press, OxfordMATH

Goodearl KR (1986) Partially ordered abelian groups with interpolation. Mathematical Surveys and Monographs No. 20, American Mathematical Society, Providence

Gudder SP (1966) Uniqueness and existence properties of bounded observables. Pac J Math 19:81–93MathSciNetCrossRefMATH

Halmos PR (1974) Measure theory. Springer, BerlinMATH

Jenča G (2010) Sharp and meager elements in orthocomlete homogeneous effect algebras. Order 27:41–61MathSciNetCrossRefMATH

Jenča G, Riečanová Z (1999) On sharp elements in lattice ordered effect algebras. Busefal 80:24–29

Jenčová A, Pulmannová S, Vinceková E (2011) Observables on \(\sigma \)-MV-algebras and \(\sigma \)-lattice effect algebras. Kybernetika 47:541–559MathSciNetMATH

Kadison R (1951) Order properties of bounded self-adjoint operators. Proc Am Math Soc 2:505–510MathSciNetCrossRefMATH

Kallenberg O (1997) Foundations of modern probability. Springer, New YorkMATH

Olson MP (1971) The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc Am Math Soc 28:537–544CrossRefMATH

Pulmannová S (2002) Compatibility and decomposition of effects. J Math Phys 43:2817–2830MathSciNetCrossRefMATH

Ravindran K (1996) On a structure theory of effect algebras, Ph.D. thesis. Kansas State University, Manhattan

Riečanová Z (2000) Generalization of blocks for d-lattice and lattice ordered effect algebras. Int J Theor Phys 39:231–237MathSciNetCrossRefMATH

Sikorski R (1964) Boolean algebras. Springer, BerlinMATH

Varadarajan VS (1968) Geometry of quantum theory, vol 1. van Nostrand, PrincetonCrossRefMATH

Title: Sum of observables on MV-effect algebras
Author: Anatolij Dvurečenskij
Publication date: 27-07-2017
Publisher: Springer Berlin Heidelberg
Published in: Soft Computing / Issue 8/2018
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-017-2741-1

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