Kybernetika 46 no. 6, 935-947, 2010

Mac Neille completion of centers and centers of Mac Neille completions of lattice effect algebras

Martin Kalina

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 subdirect product of irreducible effect algebras (\cite{R2}). 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.

Keywords:

lattice effect algebra, MacNeille completion, atom, center

Classification:

03G12, 03G27, 06B99

References:

  1. C. C. Chang: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490.   CrossRef
  2. A. Dvure\v censkij and S. Pulmannov\'a: New Trends in Quantum Structures. Kluwer Acad. Publisher, Dordrecht, Boston, London, and Isterscience, Bratislava 2000.   CrossRef
  3. D. J. Foulis and M. K. Bennett: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325-1346.   CrossRef
  4. R. J. Greechie, D. J. Foulis and S. Pulmannov\'a: The center of an effect algebra. Order 12 (1995), 91-106.   CrossRef
  5. S. P. Gudder: Sharply dominating effect algebras. Tatra Mountains Math. Publ. 15 (1998), 23-30.   CrossRef
  6. S. P. Gudder: S-dominating effect algebras. Internat. J. Theor. Phys. 37 (1998), 915-923.   CrossRef
  7. G. Jen\v ca and Z. Rie\v canov\'a: On sharp elements in lattice ordered effect algebras. BUSEFAL 80 (1999), 24-29.   CrossRef
  8. M. Kalina: On central atoms of Archimedean atomic lattice effect algebras. Kybernetika {\rm 46} (2010), 4, 609-620.   CrossRef
  9. F. K\^opka: Compatibility in D-posets. Internat. J. Theor. Phys. 34 (1995), 1525-1531.   CrossRef
  10. K. Mosn\'a: About atoms in generalized efect algebras and their effect algebraic extensions. J. Electr. Engrg. 57 (2006), 7/s, 110-113.   CrossRef
  11. K. Mosn\'a, J. Paseka and Z. Rie\v canov\'a: 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.   CrossRef
  12. J. Paseka and Z. Rie\v canov\'a: The inheritance of BDE-property in sharply dominating lattice effect algebras and $(o)$-continuous states. Soft Computing, DOI: 10.1007/s00500-010-0561-7.   CrossRef
  13. Z. Rie\v canov\'a: Compatibility and central elements in effect algebras. Tatra Mountains Math. Publ. 16 (1999), 151-158.   CrossRef
  14. Z. Rie\v canov\'a: Subalgebras, intervals and central elements of generalized effect algebras. Internat. J. Theor. Phys., 38 (1999), 3209-3220.   CrossRef
  15. Z. Rie\v canov\'a: Generalization of blocks for D-lattices and lattice ordered effect algebras. Internat. J. Theor. Phys. 39 (2000), 231-237.   CrossRef
  16. Z. Rie\v canov\'a: Orthogonal sets in effect algebras. Demontratio Mathematica 34 (2001), 525-532.   CrossRef
  17. Z. Rie\v canov\'a: Smearing of states defined on sharp elements onto effect algebras. Internat. J. Theor. Phys. 41 (2002), 1511-1524.   CrossRef
  18. Z. Rie\v canov\'a: Subdirect decompositions of lattice effect algebras. Internat. J. Theor. Phys. 42 (2003), 1425-1433.   CrossRef
  19. Z. Rie\v canov\'a: Distributive atomic effect akgebras. Demontratio Mathematica 36 (2003), 247-259.   CrossRef
  20. Z. Rie\v canov\'a: Lattice effect algebras densely embeddable into complete ones. Kybernetika, to appear.   CrossRef
  21. Z. Rie\v canov\'a and I. Marinov\'a: Generalized homogenous, prelattice and MV-effect algebras. Kybernetika 41 (2005), 129-142.   CrossRef