# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Adamski, W., Extensions of tight set functions with applications in topological measure theory, Trans. amer. math. soc., 283, 1, 353-368, (1984) · Zbl 0508.28001  Bova, S.; Flaminio, T., The coherence of łukasiewicz assessments is NP-complete, Int. J. approx. reason., 51, 3, 294-304, (2010) · Zbl 1201.68117  Cignoli, R.L.O.; D’Ottaviano, I.M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, Trends in logic—studia logica library, vol. 7, (2000), Kluwer Academic Publishers Dordrecht  Ewald, G., Combinatorial convexity and algebraic geometry, vol. 168, (1996), Springer-Verlag New York · Zbl 0869.52001  Flaminio, T.; Montagna, F., MV-algebras with internal states and probabilistic fuzzy logics, Int. J. approx. reason., 50, 1, 138-152, (2009) · Zbl 1185.06007  Gierz, G.; Hofmann, K.H.; Keimel, K.; Lawson, J.D.; Mislove, M.; Scott, D.S., Continuous lattices and domains, Encyclopedia of mathematics and its applications, vol. 93, (2003), Cambridge University Press Cambridge  Halmos, P.R., Measure theory, (1950), D. Van Nostrand · Zbl 0117.10502  Keimel, K.; Lawson, J.D., Measure extension theorems for T_{0}-spaces, Topology appl., 149, 1-3, 57-83, (2005) · Zbl 1152.06301  Kelley, J.L., General topology, (1955), D. Van Nostrand Company, Inc. Toronto-New York-London · Zbl 0066.16604  Kisyński, J., On the generation of tight measures, Stud. math., 30, 141-151, (1968) · Zbl 0157.37301  Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, Trends in logic—studia logica library, vol. 8, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002  Kroupa, T., Every state on semisimple MV-algebra is integral, Fuzzy sets and systems, 157, 20, 2771-2782, (2006) · Zbl 1107.06007  V. Marra, Is there a probability theory ofmany-valued events? Probability, Uncertainty and Rationality, vol. 10, CRMSeries, Ed.Norm., Pisa (2010) 141-166. · Zbl 1206.03024  McNaughton, R., A theorem about infinite-valued sentential logic, J. symb. logic, 16, 1-13, (1951) · Zbl 0043.00901  Mundici, D., Averaging the truth-value in łukasiewicz logic, Stud. log., 55, 1, 113-127, (1995) · Zbl 0836.03016  Mundici, D., Bookmaking over infinite-valued events, Int. J. approx. reason., 43, 3, 223-240, (2006) · Zbl 1123.03011  Mundici, D., The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete contin. dyn. syst., 21, 2, 537-549, (2008) · Zbl 1154.28007  Mundici, D., A dvanced łukasiewicz calculus and MV-algebras, Trends in logic, vol. 35, (2011), Springer  Navara, M., Triangular norms and measures of fuzzy sets, (), 345-390 · Zbl 1073.28015  Panti, G., Invariant measures in free MV-algebras, Communications in algebra, 36, 8, 2849-2861, (2008) · Zbl 1154.06008  Rao, M.M., Measure theory and integration. pure and applied mathematics (New York), (1987), John Wiley & Sons, Inc./A Wiley-Interscience Publication New York  Rudin, W., Functional analysis, Mcgraw-Hill series in higher mathematics, (1973), McGraw-Hill Book Co. New York · Zbl 0253.46001  Topsøe, F., Topology and measure, Lecture notes in mathematics, vol. 133, (1970), Springer-Verlag Berlin · Zbl 0197.33301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.