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The inheritance of BDE-property in sharply dominating lattice effect algebras and \((o)\)-continuous states. (English) Zbl 1247.03135
An effect algebra is a partial algebra with a commutative partial operation \(+\) of addition, which was introduced by D. J. Foulis and M. K. Bennett [Found. Phys. 24, No. 10, 1331–1352 (1994; Zbl 1213.06004)] and can model quantum-mechanical measurements.
The main subject of the paper under review is some important substructures of Archimedean atomic lattice effect algebras like blocks, the set of sharp elements, center, etc. The main results are: (i) For every sharply dominating Archimedean atomic lattice effect algebra, every atomic block is gain sharply dominating, and every of its elements can be decomposed, i.e., it has the so-called basic decomposition property (BDP). (ii) The state smearing theorem for compactly generated Archimedean atomic lattice effect algebras is proved.

03G12 Quantum logic
Full Text: DOI
[1] Cignoli RLO, D’Ottaviano ML, Mundici D (2000) Algebraic foundations of manyvalued reasoning. Kluwer, Dordrecht
[2] Di Nola A, Dvurečenskij A (2009a) On some classes of state-morphism MV-algebras. Math Slovaca 59:517–534. doi: 10.2478/s12175-009-0145-0 · Zbl 1212.06028
[3] Di Nola A, Dvurečenskij A (2009b) State-morphism MV-algebras. Ann Pure Appl Logic 161:161–173. doi: 10.1016/j.apal.2009.05.003 · Zbl 1186.06007
[4] Dvurečenskij A (2009) On states on MV-algebras and their applications. J Logic Comput. doi: 10.1093/logcom/exp011
[5] Flaminio T, Montagna F (2007) An algebraic approach to states on MV-algebras. In: Novák V (ed) Fuzzy logic 2, Proceedings of the 5th EUSFLAT conference, 11–14 September, vol. II:201–206
[6] Flaminio T, Montagna F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50:138–152 · Zbl 1185.06007
[7] Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1325–1346
[8] Greechie RJ (1971) Orthomodular lattices admitting no states. J Combin Theor A 10:119–132 · Zbl 0219.06007
[9] Greechie RJ, Foulis DJ, Pulmannová S (1995) The center of an effect algebra. Order 12:91–106 · Zbl 0846.03031
[10] Gudder SP (1998a) Sharply dominating effect algebras. Tatra Mt Math Publ 15:23–30 · Zbl 0939.03073
[11] Gudder SP (1998b) S-dominating effect algebras. Int J Theor Phys 37:915–923 · Zbl 0932.03072
[12] Jenča G, Riečanová Z (1999) On sharp elements in lattice ordered effect algebras. BUSEFAL 80:24–29
[13] Jenča G, Pulmannová S (2003) Orthocomplete effect algebras. Proc Am Math Soc 131:2663–2671 · Zbl 1019.03046
[14] Kalmbach G (1983) Orthomodular lattices. Kluwer, Dordrecht · Zbl 0512.06011
[15] Kôpka F (1995) Compatibility in D-posets. Int J Theor Phys 34:1525–1531 · Zbl 0843.03042
[16] Kroupa T (2006) Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst 157:2771–2782 · Zbl 1107.06007
[17] Kühr J, Mundici D (2007) De Finetti theorem and Borel states in [0, 1]-valued algebraic logic. Int J Approx Reason 46:605–616 · Zbl 1189.03076
[18] Mosná K (2007) Atomic lattice effect algebras and their sub-lattice effect algebras. J Elect Eng 58(7/S):3–6
[19] Olejček V (2007) An atomic MV-effect algebra with non-atomic center. Kybernetika 43:343–346 · Zbl 1149.06006
[20] Paseka J, Riečanová Z (2009) Compactly generated de Morgan lattices, basic algebras and effect algebras. Int J Theor Phys. doi: 10.1007/s10773-009-0011-4
[21] Paseka J, Riečanová Z, Wu J (2009) Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras, preprint, http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.3288v2.pdf
[22] Riečanová Z (1992) Measures and topologies on atomic quantum logics. In: Mathematical research. Topology measures and fractals, vol 66. Akademie Verlag, pp 154–160 · Zbl 0777.06008
[23] Riečanová Z (1999a) Compatibility and central elements in effect algebras. Tatra Mountains Math Publ 16:151–158 · Zbl 0949.03063
[24] Riečanová Z (1999b) Subalgebras, intervals and central elements of generalized effect algebras. Int J Theor Phys 38:3209–3220 · Zbl 0963.03087
[25] Riečanová Z (2000) Generalization of blocks for D-lattices and lattice-ordered effect algebras. Int J Theor Phys 39:231–237 · Zbl 0968.81003
[26] Riečanová Z (2001a) Orthogonal sets in effect algebras. Demons Math 34(3):525–532 · Zbl 0989.03071
[27] Riečanová Z (2001b) Proper effect algebras admitting no states. Int J Theor Phys 40:1683–1691 · Zbl 0989.81003
[28] Riečanová Z (2002a) Smearings of states defined on sharp elements onto effect algebras. Int J Theor Phys 41:1511–1524 · Zbl 1016.81005
[29] Riečanová Z (2002b) States, uniformities and metrics on lattice effect algebras. Int J Uncertain Fuzziness Knowl Based Syst 10:125–133 · Zbl 1059.03079
[30] Riečanová Z (2003a) Continuous lattice effect algebras admitting order-continuous states. Fuzzy Sets Syst 136:41–54 · Zbl 1022.03047
[31] Riečanová Z (2003b) Subdirect decompositions of lattice effect algebras. Int J Theor Phys 42:1415–1423
[32] Riečanová Z (2005) Basic decomposition of elements and Jauch–Piron effect algebras. Fuzzy Sets Syst 155:138–149 · Zbl 1073.81014
[33] Riečanová Z (2006) Archimedean atomic lattice effect algebras in which all sharp elements are central. Kybernetika 42:143–150 · Zbl 1249.03121
[34] Riečanová Z, Wu J (2008) States on sharply dominating effect algebras. Sci China Ser A Math 51:907–914 · Zbl 1155.81012
[35] Riečanová Z (2009) Pseudocomplemented lattice effect algebras and existence of states. Inform Sci 179:529–534 · Zbl 1166.03038
[36] Riečanová Z, Paseka J (2009) State smearing theorems and the existence of states on some atomic lattice effect algebras. J Logic Comput Adv Access. doi: 10.1093/logcom/exp018
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