Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T05:57:20.299Z Has data issue: false hasContentIssue false
  • English
  • Français

The block structure of complete lattice ordered effect algebras

Part of: Algebraic logic Modular lattices, complemented lattices

Published online by Cambridge University Press:  09 April 2009

Gejza Jenča
Gejza Jenča
Affiliation:
Department of Mathematics Faculty of Electrical Engineering and Information TechnologyIlkovičova 3812 19 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that every for every complete lattice-ordered effect algebra E there exists an orthomodular lattice O(E) and a surjective full morphism øE: O(E) → E which preserves blocks in both directions: the (pre)imageofa block is always a block. Moreover, there is a 0, 1-lattice embedding : E → O(E).

MSC classification

Secondary: 06C15: Complemented lattices, orthocomplemented lattices and posets 03G12: Quantum logic 81P10: Logical foundations of quantum mechanics; quantum logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 2 , October 2007 , pp. 181 - 216
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bennett, M. K. and Foulis, D. J., ‘Phi-symmetric effect algebras’, Found. Phys. 25 (1995), 1699–1722.CrossRefGoogle Scholar
[2]Bennett, M. K. and Foulis, D. J., ‘Interval and scale effect algebras’, Adv. in Appl. Math. 19 (1997), 200–215.Google Scholar
[3]Cattaneo, G., ‘A unified framework for the algebra of unsharp quantum mechanics’, Internat. J. Theoret. Phys. 36 (1997), 3085–3117.CrossRefGoogle Scholar
[4]Chang, C. C., ‘Algebraic analysis of many-valued logics’, Trans. Amer. Math. Soc. 89 (1959), 74–80.Google Scholar
[5]Chovanec, F. and Kôpka, F., ‘Boolean D-posets’, Tatra Mt. Math. Publ 10 (1997), 1–15.Google Scholar
[6]Dvurečenskij, A. and Pulmannova, S. ’, ‘Tensor product of D-posets and D-test spaces’, Rep. Math. Phys. 34 (1994), 251–275.Google Scholar
[7]Dvurečenskij, A. and Pulmannová, S., New Trends in Quantum Structures (Kluwer, Dordrecht and Ister Science, Bratislava, 2000).Google Scholar
[8]Foulis, D. and Bennett, M. K., ‘Tensor products of orthoalgebras’, Order 10 (1993), 271–282.Google Scholar
[9]Foulis, D. J. and Bennett, M. K., ‘Effect algebras and unsharp quantum logics’, Found. Phys. 24 (1994), 1325–1346.Google Scholar
[10]Foulis, D. J., Greechie, R. and Rütimann, G., ‘Filters and supports in orthoalgebras’, Internat. J. Theoret. Phys. 35 (1995), 789–802.Google Scholar
[11]Foulis, D. J. and Randall, C. H., ‘Operational quantum statistics. I. Basic concepts’, J. Math. Phys. 13 (1972), 1667–1675.Google Scholar
[12]Giuntini, R. and Greuling, H., ‘Toward a formal language for unsharp properties’, Found. Phys. 19 (1989), 931–945.CrossRefGoogle Scholar
[13]Grätzer, G., General Lattice Theory, second edition (Birkhäuser, 1998).Google Scholar
[14]Greechie, R., Foulis, D. and Pulmannová, S., ‘The center of an effect algebra’, Order 12 (1995), 91–106.CrossRefGoogle Scholar
[15]Gudder, S., ‘S-dominating effect algebras’, Internat. J. Theoret. Phys. 37 (1998), 915–923.CrossRefGoogle Scholar
[16]Gudder, S., ‘Sharply dominating effect algebras’, Tatra Mt. Math.Publ. 15 (1998), 23–30.Google Scholar
[17]Hashimoto, J., ‘Ideal theory for lattices’, Math. Japon. 2 (1952), 149–186.Google Scholar
[18]Jenča, G., ‘Blocks of homogeneous effect algebras’, Bull. Austral.Math. Soc. 64 (2001), 81–98.Google Scholar
[19]Jenča, G., ‘Finite homogeneous and lattice ordered effect algebras’, Discrete Math. 272 (2003), 197–214.Google Scholar
[20]Jenča, G., ‘Boolean algebras R-generated by MV-effect algebras’, Fuzzy sets and systems 145 (2004), 279–285.Google Scholar
[21]Jenča, G., ‘Sharp and meager elements in orthocomplete homogeneous effect algebras’, Technical report, (FEI STU Bratislava, 2004).Google Scholar
[22]Jenča, G. and Pulmannová, S., ‘Quotients of partial abelian monoids and the Riesz decomposition property’, Algebra Universalis 47 (2002), 443–177.Google Scholar
[23]Jenča, G., ‘Orthocomplete effect algebras’, Proc. Amer.Math. Soc. 131 (2003), 2663–2671.Google Scholar
[24]Jenča, G. and Riečanová, Z., ‘On sharp elements in lattice ordered effect algebras’, BUSEFAL 80 (1999), 24–29.Google Scholar
[25]Kôpka, F., ‘D-posets of fuzzy sets’, Tatra Mt. Math. Publ. 1 (1992), 83–87.Google Scholar
[26]Kôpka, F. and Chovanec, F., ‘D-posets’, Math. Slovaca 44 (1994), 21–34.Google Scholar
[27]MacNeille, H. M., ‘Extension of a distributive lattice to a Boolean ring’, Bull. Amer. Math. Soc. 45 (1939), 452–455.Google Scholar
[28]Mundici, D., ‘Interpretation of AF C*-algebras in Lukasziewicz sentential calculus’, J. Fund. Anal. 65 (1986), 15–63.Google Scholar
[29]Randall, C. H. and Foulis, D. J., ‘Operational quantum statistics. II. Manual of operations and their logics’, J. Math. Phys. 13 (1972), 1667–1675.Google Scholar
[30]Riečanová, Z., ‘A generalization of blocks for lattice effect algebras’, Internal J. Theoret. Phys. 39 (2000), 855–865.Google Scholar
[31]Riečanová, Z., ‘Continuous lattice effect algebras admitting order continuous states’, Fuzzy sets and systems 136 (2003), 41–54.CrossRefGoogle Scholar