Kybernetika 51 no. 5, 739-746, 2015

Remarks on effect-tribes

Sylvia Pulmannová and Elena VincekováDOI: 10.14736/kyb-2015-5-0739


We show that an effect tribe of fuzzy sets ${\mathcal T}\subseteq [0,1]^X$ with the property that every $f\in {\mathcal T}$ is ${\mathcal B}_0({\mathcal T})$-measurable, where ${\mathcal B}_0({\mathcal T})$ is the family of subsets of $X$ whose characteristic functions are central elements in ${\mathcal T}$, is a tribe. Moreover, a monotone $\sigma$-complete effect algebra with RDP with a Loomis-Sikorski representation $(X, {\mathcal T},h)$, where the tribe ${\mathcal T}$ has the property that every $f\in {\mathcal T}$ is ${\mathcal B}_0({\mathcal T})$-measurable, is a $\sigma$-MV-algebra.


Riesz decomposition property, MV-algebra, tribe, effect-tribe, monotone $\sigma $-complete effect algebra


81P10, 81P15


  1. D. Buhagiar, E. Chetcuti and A. Dvurečenskij: Loomis-Sikorski representation of monotone $\sigma$-complete effect algebras. Fuzzy Sets Syst. 157 (2006), 683-690.   DOI:10.1016/j.fss.2005.09.013
  2. D. Butnariu and E. P. Klement: Triangular Norm Based Measures and Games with Fuzzy Coalitions. Kluwer Academic Publisher, Dordrecht 1993.   DOI:10.1007/978-94-017-3602-2
  3. A. Dvurečenskij: Representation of states on effect-tribes and effect algebras by integrals. Rep. Math. Phys. 67 (2011), 63-85.   DOI:10.1016/s0034-4877(11)80011-x
  4. A. Dvurečenskij: Smearing of observables and spectral measures on quantum structures. Found. Phys. 43 (2013), 210-224.   DOI:10.1007/s10701-012-9689-x
  5. A. Dvurečenskij: Central elements and Cantor-Bernstein's theorem for pseudo effect algebras. J. Austral. Math. Soc. 74 (2003), 121-143.   DOI:10.1017/s1446788700003177
  6. A. Dvurečenskij: Loomis-Sikorski theorem for $\sigma$-complete MV-algebras and $\ell$-groups. J. Austral. Math. Soc. Ser. A 68 (2000), 261-277.   DOI:10.1017/s1446788700001993
  7. A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structures. Kluwer Academic/Ister Science, Dordrecht/Bratislava 2000.   DOI:10.1007/978-94-017-2422-7
  8. D. J. Foulis and M. K. Bennett: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325-1346.   DOI:10.1007/bf02283036
  9. R. J. Greechie, D. J. Foulis and S. Pulmannová: The center of an effect algebra. Order 12 (1995), 91-106.   DOI:10.1007/bf01108592
  10. K. R. Goodearl: Partially Ordered Abelian Groups with Interpolation. Math. Surveys and Monographs, Vol. 20, Am. Math. Soc., Providence 1986.   DOI:10.1007/bf01108592
  11. A. Jenčová, S. Pulmannová and E. Vinceková: Observables on $\sigma$-MV algebras and $\sigma$-lattice effect algebras. Kybernetika 47 (2011), 541-559.   CrossRef
  12. D. Mundici: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. Funct. Anal. 65 (1986), 15-63.   DOI:10.1016/0022-1236(86)90015-7
  13. D. Mundici: Tensor product and the Loomis-Sikorski theorem for MV-algebras. Adv. Appl. Math. 22 (1999), 227-248.   DOI:10.1006/aama.1998.0631
  14. S. Pulmannová: A spectral theorem for sigma MV-algebras. Kybernetika 41 (2005), 361-374.   CrossRef
  15. K. Ravindran: On a Structure Theory of Effect Algebras. PhD. Thesis, Kansas State Univ. Manhattan 1996.   CrossRef