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States in Łukasiewicz logic correspond to probabilities of rational polyhedra. (English) Zbl 1260.03050
A state on an MV-algebra can be seen as a probability of many-valued events (described by formulas of Łukasiewicz logic). The free MV-algebra over $$n$$ generators is an algebra of $$[0,1]$$-valued functions, called McNaughton functions. States on the free MV-algebra over $$n$$ generators coincide with integrals of the McNaughton functions with $$n$$ variables (this result applies to the more general class of semisimple MV-algebras; see [the author, Fuzzy Sets Syst. 157, No. 20, 2771–2782 (2006; Zbl 1107.06007); G. Panti, Commun. Algebra 36, No. 8, 2849–2861 (2008; Zbl 1154.06008)].
In this paper this result is strengthened, and it is shown that states over free MV-algebras over $$n$$ generators coincide with measures of rational polyhedra in $$[0,1]^n$$. Indeed, a state of a McNaughton function is completely determined by the measure of its one-set. The proof is new and elementary and does not rely on the previous results of integral representation: this makes the paper both deep in its results and pleasant (and self-contained) to read.

##### MSC:
 03B50 Many-valued logic 06D35 MV-algebras
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