MV-algebras with internal states and probabilistic fuzzy logics

In this paper we enlarge the language of MV-algebras by a unary operation @s equationally described so as to preserve the basic properties of a state in its original meaning. The resulting class of algebras will be called MV-algebras with internal state (or SMV-algebras for short). After discussing some basic algebraic properties of SMV-algebras, we apply them to the study of the coherence problem for rational assessments on many-valued events. Then we propose an algebraic treatment of the Lebesgue integral and we show that internal states defined on a divisible MV"@D-algebra can be represented by means of this more general notion of integral.

[1]  Tomás Kroupa,et al.  Every state on semisimple MV-algebra is integral , 2006, Fuzzy Sets Syst..

[2]  Daniele Mundici.  Tensor Products and the Loomis-Sikorski Theorem for MV-Algebras , 1999 .

[3]  Keith R. Pierce.  Amalgamations of lattice ordered groups , 1972 .

[4]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

[5]  Daniele Mundici,et al.  Bookmaking over infinite-valued events , 2006, Int. J. Approx. Reason..

[6]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[7]  Franco Montagna,et al.  An Algebraic Approach to States on MV-algebras , 2007, EUSFLAT Conf..

[8]  Mirko Navara,et al.  Triangular norms and measures of fuzzy sets , 2005 .

[9]  Robert McNaughton,et al.  A Theorem About Infinite-Valued Sentential Logic , 1951, J. Symb. Log..

[10]  Vincenzo Marra,et al.  The Lebesgue state of a unital abelian lattice-ordered group , 2007 .

[11]  Jeff B. Paris,et al.  A note on the Dutch Book method , 2001, ISIPTA.

[12]  DANIELE MUNDICI,et al.  Averaging the truth-value in Łukasiewicz logic , 1995, Stud Logica.

[13]  Tommaso Flaminio.  Strong non-standard completeness for fuzzy logics , 2008, Soft Comput..

[14]  D. Mundici.  Interpretation of AF -algebras in ukasiewicz sentential calculus , 1986 .

[15]  L. P. Belluce.  Semisimple Algebras of Infinite Valued Logic and Bold Fuzzy Set Theory , 1986, Canadian Journal of Mathematics.

[16]  Brunella Gerla,et al.  Many-valued Logics of Continuous t-norms and Their Functional Representation , 2001 .

[17]  Lluis Godo,et al.  A logic for reasoning about the probability of fuzzy events , 2007, Fuzzy Sets Syst..

[18]  B. M. Hill,et al.  Theory of Probability , 1990 .

[19]  M. Baaz.  Infinite-valued Gödel logics with $0$-$1$-projections and relativizations , 1996 .

[20]  Tomás Kroupa,et al.  Conditional probability on MV-algebras , 2005, Fuzzy Sets Syst..

[21]  R. Mesiar,et al.  Logical, algebraic, analytic, and probabilistic aspects of triangular norms , 2005 .

[22]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[23]  Giovanni Panti.  Invariant Measures in Free MV-Algebras , 2005, math/0508445.

[24]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[25]  Petr Hájek,et al.  Complexity of fuzzy probability logics II , 2007, Fuzzy Sets Syst..

[26]  Jan Kühr,et al.  De Finetti theorem and Borel states in [0, 1]-valued algebraic logic , 2007, Int. J. Approx. Reason..

[27]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.