Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter December 20, 2012

MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras: Generic scheme behind

  • Josef Niederle EMAIL logo and Jan Paseka
From the journal Mathematica Slovaca

Abstract

We present an example of a natural class of atomic Archimedean sharply dominating lattice effect algebras with non-bifull and atomic centers. Further, we extend and clarify known results about MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras.

[1] BLANK, J.— EXNER, P.— HAVLÍČEK, M.: Hilbert Space Operators in Quantum Physics (2nd ed.), Springer, Berlin, 2008. Search in Google Scholar

[2] DVUREČENSKIJ, A.— PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Acad. Publ./Ister Science, Dordrecht/Bratislava, 2000. 10.1007/978-94-017-2422-7Search in Google Scholar

[3] FOULIS, D. J.— BENNETT, M. K.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325–1346. http://dx.doi.org/10.1007/BF0228303610.1007/BF02283036Search in Google Scholar

[4] GREECHIE, R. J.— FOULIS, D. J.— PULMANNOVÁ, S.: The center of an effect algebra, Order 12 (1995), 91–106. http://dx.doi.org/10.1007/BF0110859210.1007/BF01108592Search in Google Scholar

[5] GUDDER, S. P.: Sharply dominating effect algebras, Tatra Mt. Math. Publ. 15 (1998), 23–30. Search in Google Scholar

[6] GUDDER, S. P.: S-dominating effect algebras, Internat. J. Theoret. Phys. 37 (1998), 915–923. http://dx.doi.org/10.1023/A:102663700113010.1023/A:1026637001130Search in Google Scholar

[7] JENČA, G.— RIEČANOVÁ, Z.: On sharp elements in lattice ordered effect algebras, BUSEFAL 80 (1999), 24–29. Search in Google Scholar

[8] KALMBACH, G.: Orthomodular Lattices, Kluwer Academic Publ., Dordrecht, 1983. Search in Google Scholar

[9] KALINA, M.: On central atoms of Archimedean atomic lattice effect algebras, Kybernetika (Prague) 46 (2010), 609–620. Search in Google Scholar

[10] KALINA, M.: MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras, Kybernetika (Prague) 46 (2010), 935–947. Search in Google Scholar

[11] KÔPKA, F.: Compatibility in D-posets, Internat. J. Theoret. Phys. 34 (1995), 1525–1531. http://dx.doi.org/10.1007/BF0067626310.1007/BF00676263Search in Google Scholar

[12] MOSNÁ, K.: Atomic lattice effect algebras and their sub-lattice effect algebras, J. Electrical Engineering 58 (2007), No. 7/S, 3–6. Search in Google Scholar

[13] NIEDERLE, J.— PASEKA, J.: More about sharp and meager elements in Archimedean atomic lattice effect algebras, Soft Comput. 16 (2012), 109–119. http://dx.doi.org/10.1007/s00500-011-0738-810.1007/s00500-011-0738-8Search in Google Scholar

[14] PASEKA, J.: Modularity, atomicity and states in Archimedean lattice effect algebras, SIGMA (003), 6 (2010), 9 pages, doi: 10.3842/SIGMA.2010.003. 10.3842/SIGMA.2010.003Search in Google Scholar

[15] PASEKA, J.— RIEČANOVÁ, Z.— JUNDE, WU: Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras, Kybernetika (Prague) 46 (2010), 953–970. Search in Google Scholar

[16] PASEKA, J.— RIEČANOVÁ, Z.: The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states, Soft Comput. 15 (2011), 543–555. http://dx.doi.org/10.1007/s00500-010-0561-710.1007/s00500-010-0561-7Search in Google Scholar

[17] RIEČANOVÁ, Z.: Compatibility and central elements in effect algebras, Tatra Mt. Math. Publ. 16 (1999), 151–158. Search in Google Scholar

[18] RIEČANOVÁ, Z.: Subalgebras, intervals and central elements of generalized effect algebras, Internat. J. Theoret. Phys. 38 (1999), 3209–3220. http://dx.doi.org/10.1023/A:102668221576510.1023/A:1026682215765Search in Google Scholar

[19] RIEČANOVÁ, Z.: Generalization of blocks for D-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys. 39 (2000), 231–237. http://dx.doi.org/10.1023/A:100361980602410.1023/A:1003619806024Search in Google Scholar

[20] RIEČANOVÁ, Z.: Orthogonal sets in effect algebras, Demonstratio Math. 34 (2001), 525–532. Search in Google Scholar

[21] RIEČANOVÁ, Z.: Proper effect algebras admitting no states, Internat. J. Theoret. Phys. 40 (2001), 1683–1691. http://dx.doi.org/10.1023/A:101191151241610.1023/A:1011911512416Search in Google Scholar

[22] RIEČANOVÁ, Z.— JUNDE, WU: States on sharply dominating effect algebras, Sci. China Ser. A 51 (2008), 907–914. http://dx.doi.org/10.1007/s11425-007-0163-810.1007/s11425-007-0163-8Search in Google Scholar

[23] RIEČANOVÁ, Z.— PASEKA, J.: State smearing theorems and the existence of states on some atomic lattice effect algebras, J. LogicC omput. 21 (2011), 863–882. http://dx.doi.org/10.1093/logcom/exp01810.1093/logcom/exp018Search in Google Scholar

[24] SCHMIDT, J.: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hülle einer Geordneten Menge, Arch. Math. (Basel) 7 (1956), 241–249. http://dx.doi.org/10.1007/BF0190029710.1007/BF01900297Search in Google Scholar

[25] VARADARAJAN, V. S.: The Geometry of Quantum Mechanics, Springer-Verlag, New York, 1985. Search in Google Scholar

Published Online: 2012-12-20
Published in Print: 2012-12-1

© 2012 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 2.10.2023 from https://www.degruyter.com/document/doi/10.2478/s12175-012-0073-2/html
Scroll to top button