States in Łukasiewicz logic correspond to probabilities of rational polyhedra
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); \textit{G. Panti}, Commun. Algebra 36, No. 8, 2849--2861 (2008; Zbl 1154.06008)].NEWLINENEWLINEIn 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.
scientific article; zbMATH DE number 6080889
scientific article