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MV-algebras with internal states and probabilistic fuzzy logics. (English) Zbl 1185.06007
Summary: We enlarge the language of MV-algebras by a unary operation \(\sigma \) 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\(_{\Delta }\)-algebra can be represented by means of this more general notion of integral.

MSC:
06D35 MV-algebras
03B52 Fuzzy logic; logic of vagueness
28A99 Classical measure theory
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