| Title:
|
Atomicity of lattice effect algebras and their sub-lattice effect algebras (English) |
| Author:
|
Paseka, Jan |
| Author:
|
Riečanová, Zdenka |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
45 |
| Issue:
|
6 |
| Year:
|
2009 |
| Pages:
|
1040-1051 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show some families of lattice effect algebras (a common generalization of orthomodular lattices and MV-effect algebras) each element E of which has atomic center C(E) or the subset S(E) of all sharp elements, resp. the center of compatibility B(E) or every block M of E. The atomicity of E or its sub-lattice effect algebras C(E), S(E), B(E) and blocks M of E is very useful equipment for the investigations of its algebraic and topological properties, the existence or smearing of states on E, questions about isomorphisms and so. Namely we touch the families of complete lattice effect algebras, or lattice effect algebras with finitely many blocks, or complete atomic lattice effect algebra E with Hausdorff interval topology. (English) |
| Keyword:
|
non-classical logics |
| Keyword:
|
D-posets |
| Keyword:
|
effect algebras |
| Keyword:
|
MV-algebras |
| Keyword:
|
atomicity |
| MSC:
|
03G12 |
| MSC:
|
06D35 |
| MSC:
|
06F25 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 1252.06007 |
| idMR:
|
MR2650082 |
| . |
| Date available:
|
2010-06-02T19:35:29Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140025 |
| . |
| Reference:
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[1] E. G. Beltrametti and G. Cassinelli: The Logic of Quantum Mechanics.Addison-Wesley, Reading, MA 1981. MR 0635780 |
| Reference:
|
[2] C. C. Chang: Algebraic analysis of many-valued logics.Trans. Amer. Math. Soc. 88 (1958), 467–490. Zbl 0084.00704, MR 0094302 |
| Reference:
|
[3] F. Chovanec and F. Kôpka: Difference posets in the quantum structures background.Internat. J. Theoret. Phys. 39 (2000), 571–583. MR 1790895 |
| Reference:
|
[4] D. J. Foulis and M. K. Bennett: Effect algebras and unsharp quantum logics.Found. Phys. 24 (1994), 1325–1346. MR 1304942 |
| Reference:
|
[5] R. J. Greechie, D. J. Foulis, and S. Pulmannová: The center of an effect algebra.Order 12 (1995), 91–106. MR 1336539 |
| Reference:
|
[6] S. P. Gudder: Sharply dominating effect algebras.Tatra Mt. Math. Publ. 15 (1998), 23–30. Zbl 0939.03073, MR 1655076 |
| Reference:
|
[7] S. P. Gudder: S-dominating effect algebras.Internat. J. Theoret. Phys. 37 (1998), 915–923. Zbl 0932.03072, MR 1624277 |
| Reference:
|
[8] G. Jenča and Z. Riečanová: On sharp elements in lattice ordered effect algebras.BUSEFAL 80 (1999), 24–29. |
| Reference:
|
[9] G. Jenča and S. Pulmannová: Orthocomplete effect algebras.Proc. Amer. Math. Soc. 131 (2003), 2663–2671. MR 1974321 |
| Reference:
|
[10] G. Kalmbach: Orthomodular Lattices.Kluwer Academic Publishers, Dordrecht 1998. Zbl 0554.06009 |
| Reference:
|
[11] M. Katětov: Remarks on Boolean algebras.Colloq. Math. 11 (1951), 229–235. MR 0049862 |
| Reference:
|
[12] F. Kôpka: Compatibility in D-posets.Internat. J. Theor. Phys. 34 (1995), 1525–1531. MR 1353696 |
| Reference:
|
[13] K. Mosná: Atomic lattice effect algebras and their sub-lattice effect algebras.J. Electr. Engrg. 58 (2007), 7/S, 3–6. |
| Reference:
|
[14] V. Olejček: An atomic MV-effect algebra with non-atomic center.Kybernetika 43 (2007), 343–346. MR 2362723 |
| Reference:
|
[15] J. Paseka and Z. Riečanová: Isomorphism theorems on generalized effect algebras based on atoms.Inform. Sci. 179 (2009), 521–528. MR 2490192 |
| Reference:
|
[16] J. Paseka and Z. Riečanová: Compactly generated de Morgan lattices, basic algebras and effect algebras.Internat. J. Theoret. Phys. (2009), doi:10.1007/s10773-009-0011-4. MR 2738081 |
| Reference:
|
[17] S. Pulmannová and Z. Riečanová: Block-finite atomic orthomodular lattices.J. Pure Appl. Algebra 89 (1993), 295–304. |
| Reference:
|
[18] Z. Riečanová: Lattices and quantum logics with separated intervals, atomicity.Internat. J. Theoret. Phys. 37 (1998), 191–197. MR 1637165 |
| Reference:
|
[19] Z. Riečanová: Compatibility and central elements in effect algebras.Tatra Mt. Math. Publ. 16 (1999), 151–158. MR 1725293 |
| Reference:
|
[20] Z. Riečanová: Subalgebras, intervals and central elements of generalized effect algebras.Internat. J. Theoret. Phys. 38(1999), 3209–3220. MR 1764459 |
| Reference:
|
[21] Z. Riečanová: Archimedean and block-finite lattice effect algebras.Demonstratio Math. 33 (2000), 443–452. MR 1791464 |
| Reference:
|
[22] Z. Riečanová: Generalization of blocks for D-lattices and lattice-ordered effect algebras.Internat. J. Theoret. Phys. 39 (2000), 231–237. MR 1762594 |
| Reference:
|
[23] Z. Riečanová: Orthogonal sets in effect algebras.Demonstratio Math. 34 (2001), 3, 525–532. MR 1853730 |
| Reference:
|
[24] Z. Riečanová: Smearings of states defined on sharp elements onto effect algebras.Internat. J. Theoret. Phys. 41 (2002), 1511–1524. MR 1932844 |
| Reference:
|
[25] Z. Riečanová: Distributive atomic effect algebras.Demonstratio Math. 36 (2003), 247–259. MR 1984337 |
| Reference:
|
[26] Z. Riečanová: Continuous lattice effect algebras admitting order-continuous states.Fuzzy Sests and Systems 136 (2003), 41–54. MR 1978468 |
| Reference:
|
[27] Z. Riečanová: Subdirect decompositions of lattice effect algebras.Internat. J. Theoret. Phys. 42 (2003), 1415–1423. MR 2021221 |
| Reference:
|
[28] Z. Riečanová: Modular atomic effect algebras and the existence of subadditive states.Kybernetika 40 (2004), 459–468. MR 2102364 |
| Reference:
|
[29] Z. Riečanová: Basic decomposition of elements and Jauch–Piron effect algebras.Fuzzy Sets and Systems 155 (2005), 138–149. MR 2206659 |
| Reference:
|
[30] Z. Riečanová: Archimedean atomic lattice effect algebras in which all sharp elements are central.Kybernetika 42 (2006), 143–150. MR 2241781 |
| Reference:
|
[31] Z. Riečanová and Wu Junde: States on sharply dominating effect algebras.Sci. China Ser. A: Mathematics 51 (2008), 907–914. MR 2395393 |
| Reference:
|
[32] Z. Riečanová: Pseudocomplemented lattice effect algebras and existence of states.Inform. Sci. 179 (2009), 529–534. MR 2490193 |
| Reference:
|
[33] Z. Riečanová and J. Paseka: State smearing theorems and the existence of states on some atomic lattice effect algebras.J. Logic and Computation, Advance Access, published on March 13, 2009, doi:10.1093/logcom/exp018. |
| Reference:
|
[34] T. A. Sarymsakov, S. A. Ajupov, Z. Chadzhijev, and V. J. Chilin: Ordered Algebras.(in Russian) FAN, Tashkent, 1983. MR 0781349 |
| Reference:
|
[35] J. Schmidt: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hülle einer Geordneten Menge.Arch. Math. 7 (1956), 241–249. MR 0084484 |
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