Kybernetika 47 no. 1, 100-109, 2011

Lattice effect algebras densely embeddable into complete ones

Zdenka Riečanová

Abstract:

An effect algebraic partial binary operation $\oplus$ defined on the underlying set $E$ uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion $\widehat{E}$ of $E$ there exists an effect algebraic partial binary operation $\widehat{\oplus}$ then $\widehat{\oplus}$ need not be an extension of ${\oplus}$. Moreover, for an Archimedean atomic lattice effect algebra $E$ we give a necessary and sufficient condition for that $\widehat{\oplus}$ existing on $\widehat{E}$ is an extension of ${\oplus}$ defined on $E$. Further we show that such $\widehat{\oplus}$ extending ${\oplus}$ exists at most one.

Keywords:

non-classical logics, effect algebras, $MV$-algebras, orthomodular lattices, MacNeille completions

Classification:

03G12, 06D35, 06F25, 81P10

paper.pdf

References:

  1. C. C.~Chang: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958). 467-490.   CrossRef
  2. F. Chovanec and F. K\^opka: Difference posets in the quantum structures background. Internat. J. Theoret. Phys. 39 (2000), 571-583.   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 Mt. Math. Publ. 15 (1998), 23-30.   CrossRef
  6. S. P. Gudder: S-dominating effect algebras. Internat. J. Theoret. Phys. 37 (1998), 915-923.   CrossRef
  7. G. Jen\v{c}a and Z. Rie\v{c}anov\'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 46 (2010), 609-620.   CrossRef
  9. M. Kalina, V. Olej\v cek, J. Paseka and Z. Rie\v c{}anov\'a: Sharply dominating $MV$-effect algebras. {\it To appear in}: Internat. J. Theoret. Phys. DOI: 10.1007/s10773-010-0338-x.   CrossRef
  10. G. Kalmbach: Orthomodular Lattices. Kluwer Academic Publ. Dordrecht 1998.   CrossRef
  11. F. K\^opka: Compatibility in D-posets. Internat. J. Theoret. Phys. 34 (1995), 1525-1531.   CrossRef
  12. K. Mosn\'a: Atomic lattice effect algebras and their sub-lattice effect algebras. J. Electr. Engrg. 58 (2007), 7/s, 3-6.   CrossRef
  13. J. Paseka and Z. Rie\v{c}anov\'a: Isomorphism theorems on generalized effect algebras based on atoms. Inform. Sci. 179 (2009), 521-528.   CrossRef
  14. J. Paseka and Z. Rie\v canov\'a: The inheritance of BDE-property in sharply dominating lattice ef\/fect algebras and $(o)$-continuous states. {\it To appear in}: Soft Comput. DOI: 10.1007/s00500-010-0561-7.   CrossRef
  15. Z. Rie\v{c}anov\'{a}: Compatibility and central elements in effect algebras. Tatra Mountains Math. Publ. 16 (1999), 151-158.   CrossRef
  16. Z. Rie\v{c}anov\'{a}: MacNeille completions of D-posets and effect algebras. Internat. J. Theoret. Phys. 39 (2000), 859-869.   CrossRef
  17. Z. Rie\v{c}anov\'{a}: Subalgebras, intervals and central elements of generalized effect algebras. Internat. J. Theoret. Phys. 38 (1999), 3209-3220.   CrossRef
  18. Z.~Rie\v{c}anov\'a: Archimedean and block-finite lattice effect algebras. Demonstratio Math. 33 (2000), 443-452.   CrossRef
  19. Z. Rie\v{c}anov\'a: Generalization of blocks for D-lattices and lattice-ordered effect algebras. Internat. Jour. Theoret. Phys. 39 (2000), 231-237.   CrossRef
  20. Z. Rie\v canov\'a: Orthogonal sets in effect algebras. Demonstratio Math. 34 (2001), 3, 525-532.   CrossRef
  21. Z. Rie\v{c}anov\'a: Smearings of states defined on sharp elements onto effect algebras. Internat. J. Theoret. Phys. 41 (2002), 1511-1524.   CrossRef
  22. Z. Rie\v{c}anov\'a: Distributive atomic effect algebras. Demonstratio Math. 36 (2003), 247-259.   CrossRef
  23. Z. Rie\v{c}anov\'a: Subdirect decompositions of lattice effect algebras. Internat. J. Theoret. Phys. 42 (2003), 1415-1423.   CrossRef
  24. Z. Rie\v{c}anov\'a: Pseudocomplemented lattice effect algebras and existence of states. Inform. Sci. 179 (2009) 529-534.   CrossRef
  25. Z. Rie\v{c}anov\'a: Archimedean atomic lattice effect algebras with complete lattice of sharp elements. SIGMA 6 (2010), 001, 8 pages.   CrossRef