The MV-Algebraic Loomis–Sikorski Theorem

Chapter
Part of the Trends in Logic book series (TREN, volume 35)

Abstract

An MV-algebra A is said to be σ-complete if its underlying lattice is closed under countable suprema. It follows that A is semisimple, whence it is isomorphic to an MV-algebra A* of continuous [0,1]-valued functions defined on some compact Hausdorff space X.

Keywords

Boolean Algebra Compact Hausdorff Space Countable Sequence Nonempty Closed Subset Clopen Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Cignoli, R. L. O., D’Ottaviano, I. M. L., Mundici, D. (2000). Algebraic foundations of many-valued reasoning, Volume 7 of Trends in Logic. Dordrecht: Kluwer.Google Scholar
  2. 2.
    Riečan, B., Neubrunn, T. (1997). Integral, Measure, and Ordering. Dordrecht: Kluwer.MATHGoogle Scholar
  3. 3.
    Klement, E. P., Mesiar, R., Pap, E. (2000). Triangular norms, Trends in Logic, Vol. 8. Dordrecht: Kluwer.Google Scholar
  4. 4.
    Stone, M. H. (1949). Boundedness properties in function-lattices. Canadian Journal of Mathematics, 1.2, 176–186.CrossRefMATHGoogle Scholar
  5. 5.
    Nakano, H. (1941). Über das System aller stetigen Funktionen auf einem topologischen Raum. Proceedings of the Imperial Academy. Tokyo, 17, 308–310.CrossRefMATHGoogle Scholar
  6. 6.
    Sikorski, R. (1960). Boolean Algebras. Ergebnisse Math. Grenzgeb. Berlin: Springer.MATHGoogle Scholar
  7. 7.
    Mundici, D. (1999). Tensor products and the Loomis–Sikorski theorem for MV-algebras, Advances in Applied Mathematics, 22, 227–248.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dvurečenskij, A. (2000). Loomis–Sikorski theorem for \(\sigma\)-complete MV-algebras and \(\ell\)-groups, Journal of the Australian Mathematical Society, 68, 261–277.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Butnariu, D., Klement, E. P. (1995). Triangular norm-based measures and games with fuzzy coalitions. Dordrecht: Kluwer.Google Scholar
  10. 10.
    Cignoli, R., Mundici, D. (2006). Stone duality for Dedekind \(\sigma\)-complete \(\ell\)-groups with order-unit. Journal of Algebra, 302, 848–861.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics “Ulisse Dini”University of FlorenceFlorenceItaly

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