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Rota, Probability, Algebra and Logic

Invited Chapter
  • Daniele Mundici

Abstract

Inspired by Rota’s Fubini Lectures, we present the MV-algebraic extensions of various results in probability theory, first proved for boolean algebras by De Finetti, Kolmogorov, Carathéodory, Loomis, Sikorski and others. MV-algebras stand to Łukasiewicz infinite-valued logic as boolean algebras stand to boolean logic. Using Elliott’s classification, the correspondence between countable boolean algebras and commutative AF C*-algebras extends to a correspondence between countable MV-algebras and AF C*-algebras whose Murray-von Neumann order of projections is a lattice. In this way, (faithful, invariant) MV-algebraic states are identified with (faithful, invariant) tracial states of their corresponding AF C*-algebras. Faithful invariant states exist in all finitely presented MV-algebras. At the other extreme, working in the context of σ-complete MV-algebras we present a generalization of Carathéodory boolean algebraic probability theory.

Keywords

Boolean Algebra Tracial State Fuzzy Game Fuzzy Coalition Regular Borel Probability Measure 
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.

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Daniele Mundici
    • 1
  1. 1.Università di FirenzeFirenzeItaly

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