On central atoms of Archimedean atomic lattice effect algebras

Martin Kalina

Kybernetika (2010)

  • Volume: 46, Issue: 4, page 609-620
  • ISSN: 0023-5954

Abstract

top
If element z of a lattice effect algebra ( E , ⊕ , 0 , 1 ) is central, then the interval [ 0 , z ] is a lattice effect algebra with the new top element z and with inherited partial binary operation ⊕ . 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 direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether C ( E ) is a bifull sublattice of an Archimedean atomic lattice effect algebra E . We show that there exists a lattice effect algebra ( E , ⊕ , 0 , 1 ) with atomic C ( E ) which is not a bifull sublattice of E . Moreover, we show that also B ( E ) , the center of compatibility, may not be a bifull sublattice of E .

How to cite

top

Kalina, Martin. "On central atoms of Archimedean atomic lattice effect algebras." Kybernetika 46.4 (2010): 609-620. <http://eudml.org/doc/196425>.

@article{Kalina2010,
abstract = {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 direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C(E)$ is a bifull sublattice of an Archimedean atomic lattice effect algebra $E$. We show that there exists a lattice effect algebra $(E,\oplus , \{\mathbf \{0\}\}, \{\mathbf \{1\}\})$ with atomic $C(E)$ which is not a bifull sublattice of $E$. Moreover, we show that also $B(E)$, the center of compatibility, may not be a bifull sublattice of $E$.},
author = {Kalina, Martin},
journal = {Kybernetika},
keywords = {lattice effect algebra; center; atom; bifullness; orthomodular lattice; lattice effect algebra; center; atom; bifullness; state; completeness},
language = {eng},
number = {4},
pages = {609-620},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On central atoms of Archimedean atomic lattice effect algebras},
url = {http://eudml.org/doc/196425},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Kalina, Martin
TI - On central atoms of Archimedean atomic lattice effect algebras
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 4
SP - 609
EP - 620
AB - 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 direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C(E)$ is a bifull sublattice of an Archimedean atomic lattice effect algebra $E$. We show that there exists a lattice effect algebra $(E,\oplus , {\mathbf {0}}, {\mathbf {1}})$ with atomic $C(E)$ which is not a bifull sublattice of $E$. Moreover, we show that also $B(E)$, the center of compatibility, may not be a bifull sublattice of $E$.
LA - eng
KW - lattice effect algebra; center; atom; bifullness; orthomodular lattice; lattice effect algebra; center; atom; bifullness; state; completeness
UR - http://eudml.org/doc/196425
ER -

References

top
  1. Dvurečenskij, A., Pulmannová, S., New Trends in Quantum Structures, Kluwer Acad. Publisher, Dordrecht, Boston, London, and Isterscience, Bratislava 2000. (2000) MR1861369
  2. Foulis, D. J., Bennett, M. K., Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325–1346. (1994) MR1304942
  3. Greechie, R. J., Foulis, D. J., Pulmannová, S., 10.1007/BF01108592, Order 12 (1995), 91–106. (1995) MR1336539DOI10.1007/BF01108592
  4. Gudder, S. P., Sharply dominating effect algebras, Tatra Mountains Math, Publ. 15 (1998), 23–30. (1998) MR1655076
  5. Gudder, S. P., 10.1023/A:1026637001130, Internat. J. Theor. Phys. 37 (1998), 915–923. (1998) Zbl0932.03072MR1624277DOI10.1023/A:1026637001130
  6. Jenča, G., Riečanová, Z., On sharp elements in lattice ordered effect algebras, BUSEFAL 80 (1999), 24–29. (1999) 
  7. Kôpka, F., 10.1007/BF00676263, Interernat. J. Theor. Phys. 34 (1995), 1525–1531. (1995) MR1353696DOI10.1007/BF00676263
  8. Mosná, K., About atoms in generalized efect algebras and their effect algebraic extensions, J. Electr. Engrg. 57 (2006), 7/s, 110–113. (2006) 
  9. Mosná, K., Paseka, J., Riečanová, Z., Order convergence and order and interval topologies on posets and lattice effect algebras, In: UNCERTAINTY2008, Proc. Internat. Seminar, Publishing House of STU 2008, pp. 45–62. (2008) MR2395159
  10. Paseka, J., Riečanová, Z., The inheritance of BDE-property in sharply dominating lattice effect algebras and ( o ) -continuous states, Soft Computing, to appear. 
  11. Riečanová, Z., Compatibility and central elements in effect algebras, Tatra Mountains Math. Publ. 16 (1999), 151–158. (1999) MR1725293
  12. Riečanová, Z., 10.1023/A:1026682215765, Internat. J. Theor. Phys. 38 (1999), 3209–3220. (1999) MR1764459DOI10.1023/A:1026682215765
  13. Riečanová, Z., 10.1023/A:1003619806024, J. Theor. Phys. 39 (2000), 231–237. (2000) MR1762594DOI10.1023/A:1003619806024
  14. Riečanová, Z., Orthogonal sets in effect algebras, Demonstratio Math. 34 (2001), 525–532. (2001) Zbl0989.03071MR1853730
  15. Riečanová, Z., 10.1023/A:1020136531601, Interernat. J. Theor. Phys. 41 (2002), 1511–1524. (2002) MR1932844DOI10.1023/A:1020136531601
  16. Riečanová, Z., 10.1023/A:1025775827938, Interernat. J. Theor. Phys. 42 (2003), 1425–1433. (2003) Zbl1034.81003MR2021221DOI10.1023/A:1025775827938
  17. Riečanová, Z., Distributive atomic effect akgebras, Demonstratio Math. 36 (2003), 247–259. (2003) MR1984337
  18. Riečanová, Z., Marinová, I., Generalized homogenous, prelattice and MV-effect algebras, Kybernetika 41 (2005), 129–142. (2005) MR2138764

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.