MV-algebras with internal states and probabilistic fuzzy logics

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Abstract

In this paper we enlarge the language of MV-algebras by a unary operation σ 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 MVΔ-algebra can be represented by means of this more general notion of integral.

Keywords

States
MV-algebras
Coherence
Generalized Lebesgue integral
Integral representation

Cited by (0)

The present paper is an extended version of the paper “An algebraic approach to states on MV-algebras” (cf. [13]) appeared in the Proceedings of Eusflat 2007, where some of the results have already been presented.