zbMATH — the first resource for mathematics

The spectrum of the sum of observables on \(\sigma\)-complete MV-effect algebras. (English) Zbl 1415.06004
Summary: The natural question about the sum of observables on \(\sigma\)-complete MV-effect algebras, which was recently defined by A. Dvurečenskij [Int. J. Theor. Phys. 57, No. 3, 637–651 (2018; Zbl 1394.81020); Soft Comput. 22, No. 8, 2485–2493 (2018; Zbl 1398.06012)], is how it affects spectra of observables, particularly, their extremal points. We describe boundaries for extremal points of the spectrum of the sum of observables in a general case, and we give necessary and sufficient conditions under which the spectrum attains these boundary values. Moreover, we show that every bounded observable \(x\) on a complete MV-effect algebra \(E\) can be decomposed into the sum \(x=\tilde{x}+x'\), where \(\tilde{x}\) is the greatest sharp observable less than \(x\) and \(x'\) is a meager and extremally non-invertible observable.

06D35 MV-algebras
03G12 Quantum logic
Full Text: DOI
[1] Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer Acad. Publ, Dordrecht · Zbl 0937.06009
[2] Dvurečenskij, A., Representable effect algebras and observables int, J Theor Phys, 53, 2855-2866, (2014) · Zbl 1308.81011
[3] Dvurečenskij, A., Olson order of quantum observables, Int J Theor Phys, 55, 4896-4912, (2016) · Zbl 1358.81011
[4] Dvurečenskij, A., Quantum observables and effect algebras, Int J Theor Phys, (2017) · Zbl 1387.81015
[5] Dvurečenskij, A., Sum of observables on MV-effect algebras, Soft Comput, (2017) · Zbl 1398.06012
[6] Dvurečenskij, A.; Kuková, M., Observables on quantum structures, Inf Sci, 262, 215-222, (2014) · Zbl 1329.81140
[7] Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Acad. Publ, Dotrecht, p 541 + xvi · Zbl 0987.81005
[8] Foulis, DJ; Bennett, MK, Effect algebras and unsharp quantum logics, Found Phys, 24, 1331-1352, (1994) · Zbl 1213.06004
[9] Golan JS (1999) Semirings and their applications. Kluwer Acad. Publ, Dordrecht
[10] Gudder, SP, Uniqueness and existence properties of bounded observables, Pac J Math, 19, 81-93, (1966) · Zbl 0149.23603
[11] Jenča, G., Sharp and meager elements in orthocomplete homogeneous effect algebras, Order, 27, 41-61, (2010) · Zbl 1193.03084
[12] Jenča, G.; Riečanová, Z., On sharp elements in lattice ordered effect algebras, Busefal, 80, 24-29, (1999)
[13] Jenčová, A.; Pulmannová, S.; Vinceková, E., Sharp and fuzzy observables on effect algebras, Int J Theor Phys, 47, 125-148, (2008) · Zbl 1139.81006
[14] Jenčová, A.; Pulmannová, S.; Vinceková, E., Observables on \(\sigma \)-MV algebras and \(\sigma \)-lattice effect algebras, Kybernetika, 47, 541-559, (2011) · Zbl 1237.81008
[15] Kôpka, F.; Chovanec, F., D-posets, Math Slovaca, 44, 21-34, (1994) · Zbl 0789.03048
[16] Olson, MP, The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc Am Math Soc, 28, 537-544, (1971) · Zbl 0215.20504
[17] Pulmannová, S., A note on observables on MV-algebras, Soft Comput, 4, 45-48, (2000) · Zbl 1005.06006
[18] Pulmannová, S.; Vinceková, E., Compatibility of observables on effect algebras, Soft Comput, 20, 3957-3967, (2016) · Zbl 1370.03082
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.