| Title:
|
Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras (English) |
| Author:
|
Kalina, Martin |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
6 |
| Year:
|
2010 |
| Pages:
|
935-947 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
If element $z$ of a lattice effect algebra $(E,\oplus, {\mathbf 0}, {\mathbf 1})$ is central, then the interval $[{\mathbf 0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C(E)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C(E)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion $\hat{E}$ of $E$ which is its extension (i.e. $E$ is densely embeddable into $\hat{E}$) then it is possible to embed $E$ into a direct product of irreducible effect algebras. Thus $E$ inherits some of the properties of $\hat{E}$. For example, the existence of a state in $\hat{E}$ implies the existence of a state in $E$. In this context, a natural question arises if the MacNeille completion of the center of $E$ (denoted as ${\cal M}{\cal C}(C(E))$) is necessarily the same as the center of $\hat{E}$, i.e., if ${\cal M}{\cal C}(C(E))=C(\hat{E})$ is necessarily true. We show that the equality is not necessarily fulfilled. We find a necessary condition under which the equality may hold. Moreover, we show also that even the completeness of $C(E)$ and its bifullness in $E$ is not sufficient to guarantee the mentioned equality. (English) |
| Keyword:
|
lattice effect algebra |
| Keyword:
|
center |
| Keyword:
|
atom |
| Keyword:
|
MacNeille completion |
| MSC:
|
03G12 |
| MSC:
|
03G27 |
| MSC:
|
06B99 |
| idZBL:
|
Zbl 1221.06010 |
| idMR:
|
MR2797418 |
| . |
| Date available:
|
2011-04-12T12:40:32Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141457 |
| . |
| Reference:
|
[1] Chang, C. C.: Algebraic analysis of many-valued logics.Trans. Amer. Math. Soc. 88 (1958), 467–490. Zbl 0084.00704, MR 0094302, 10.1090/S0002-9947-1958-0094302-9 |
| Reference:
|
[2] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures.Kluwer Acad. Publisher, Dordrecht, Boston, London, and Isterscience, Bratislava 2000. MR 1861369 |
| Reference:
|
[3] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325–1346. MR 1304942 |
| Reference:
|
[4] Greechie, R. J., Foulis, D. J., Pulmannová, S.: The center of an effect algebra.Order 12 (1995), 91–106. MR 1336539, 10.1007/BF01108592 |
| Reference:
|
[5] Gudder, S. P.: Sharply dominating effect algebras.Tatra Mountains Math. Publ. 15 (1998), 23–30. Zbl 0939.03073, MR 1655076 |
| Reference:
|
[6] Gudder, S. P.: S-dominating effect algebras.Internat. J. Theor. Phys. 37 (1998), 915-923. Zbl 0932.03072, MR 1624277, 10.1023/A:1026637001130 |
| Reference:
|
[7] Jenča, G., Riečanová, Z.: On sharp elements in lattice ordered effect algebras.BUSEFAL 80 (1999), 24–29. |
| Reference:
|
[8] Kalina, M.: On central atoms of Archimedean atomic lattice effect algebras.Kybernetika 46 (2010), 4, 609–620. Zbl 1214.06002, MR 2722091 |
| Reference:
|
[9] Kôpka, F.: Compatibility in D-posets.Internat. J. Theor. Phys. 34 (1995), 1525–1531. MR 1353696, 10.1007/BF00676263 |
| Reference:
|
[10] Mosná, K.: About atoms in generalized efect algebras and their effect algebraic extensions.J. Electr. Engrg. 57 (2006), 7/s, 110–113. |
| Reference:
|
[11] Mosná, K., Paseka, J., Riečanová, Z.: Order convergence and order and interval topologies on posets and lattice effect algebras.In: Proc. internat. seminar UNCERTAINTY 2008, Publishing House of STU 2008, pp. 45–62. |
| Reference:
|
[12] Paseka, J., Riečanová, Z.: The inheritance of BDE-property in sharply dominating lattice effect algebras and $(o)$-continuous states.Soft Computing, DOI: 10.1007/s00500-010-0561-7. |
| Reference:
|
[13] Riečanová, Z.: Compatibility and central elements in effect algebras.Tatra Mountains Math. Publ. 16 (1999), 151–158. MR 1725293 |
| Reference:
|
[14] Riečanová, Z.: Subalgebras, intervals and central elements of generalized effect algebras.Internat. J. Theor. Phys., 38 (1999), 3209–3220. MR 1764459, 10.1023/A:1026682215765 |
| Reference:
|
[15] Riečanová, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras.Internat. J. Theor. Phys. 39 (2000), 231–237. MR 1762594, 10.1023/A:1003619806024 |
| Reference:
|
[16] Riečanová, Z.: Orthogonal sets in effect algebras.Demontratio Mathematica 34 (2001), 525–532. Zbl 0989.03071 |
| Reference:
|
[17] Riečanová, Z.: Smearing of states defined on sharp elements onto effect algebras.Internat. J. Theor. Phys. 41 (2002), 1511–1524. MR 1932844, 10.1023/A:1020136531601 |
| Reference:
|
[18] Riečanová, Z.: Subdirect decompositions of lattice effect algebras.Internat. J. Theor. Phys. 42 (2003), 1425–1433. Zbl 1034.81003, MR 2021221, 10.1023/A:1025775827938 |
| Reference:
|
[19] Riečanová, Z.: Distributive atomic effect akgebras.Demontratio Mathematica 36 (2003), 247–259. MR 1984337 |
| Reference:
|
[20] Riečanová, Z.: Lattice effect algebras densely embeddable into complete ones.Kybernetika, to appear. |
| Reference:
|
[21] Riečanová, Z., Marinová, I.: Generalized homogenous, prelattice and MV-effect algebras.Kybernetika 41 (2005), 129–142. MR 2138764 |
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