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Every state on semisimple MV-algebra is integral. (English) Zbl 1107.06007

An MV-algebra \(M\) is semisimple iff it is isomorphic to a clan of fuzzy sets, i.e., a system \({\mathcal S}\) which contains \(1\), and is closed with respect to negation and truncate sum. Moreover, the system of fuzzy sets can be chosen as a system of continuous functions on a Hausdorff compact space. Such an MV-algebra admits a separating system of states. The main result of the paper is the statement saying that every finitely additive state on a semisimple MV-algebra can be represented as an integral with respect to a Borel measure on this compact Hausdorff space.

MSC:

06D35 MV-algebras
28C99 Set functions and measures on spaces with additional structure
46A55 Convex sets in topological linear spaces; Choquet theory
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[1] Alfsen, E.M., Compact convex sets and boundary integrals, (1971), Springer New York, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57 · Zbl 0209.42601
[2] G. Barbieri, H. Weber, Measures on clans and on MV-algebras, in: Handbook of Measure Theory, vol. I, II, North-Holland, Amsterdam, 2002, pp. 911-945. · Zbl 1019.28009
[3] Belluce, L.P., Semisimple algebras of infinite valued logic and bold fuzzy set theory, Canad. J. math., 38, 6, 1356-1379, (1986) · Zbl 0625.03009
[4] Butnariu, D.; Klement, E.P., Triangular norm based measures and games with fuzzy coalitions, (1993), Kluwer Dordrecht · Zbl 0804.90145
[5] R.L.O. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Trends in Logic—Studia Logica Library, vol. 7, Kluwer Academic Publishers, Dordrecht, 2000.
[6] Dvurečenskij, A., Loomis – sikorski theorem for \(\sigma\)-complete MV-algebras and l-groups, J. austral. math. soc. ser. A, 68, 2, 261-277, (2000) · Zbl 0958.06006
[7] Dvurečenskij, A., Pseudo MV-algebras are intervals in \(l\)-groups, J. austral. math. soc., 72, 3, 427-445, (2002) · Zbl 1027.06014
[8] K.R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, 1986. · Zbl 0589.06008
[9] E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Trends in Logic—Studia Logica Library, vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
[10] E.P. Klement, M. Navara, A characterization of tribes with respect to the Łukasiewicz t-norm, Czechoslovak Math. J. 47(122) (4) (1997) 689-700. · Zbl 0902.28015
[11] Krein, M.; Milman, D., On extreme points of regular convex sets, Studia math., 9, 133-138, (1940) · Zbl 0063.03360
[12] Kroupa, T., Representation and extension of states on MV-algebras, Arch. math. logic, 45, 4, 381-392, (2006) · Zbl 1101.06008
[13] Mundici, D., Interpretation of AF \(C^*\)-algebras in łukasiewicz sentential calculus, J. funct. anal., 65, 1, 15-63, (1986) · Zbl 0597.46059
[14] Mundici, D., Averaging the truth-value in łukasiewicz logic, Studia logica, 55, 1, 113-127, (1995) · Zbl 0836.03016
[15] Mundici, D., Tensor products and the loomis – sikorski theorem for MV-algebras, Adv. appl. math., 22, 2, 227-248, (1999) · Zbl 0926.06004
[16] Navara, M., Triangular norms and measures of fuzzy sets, (), 345-390 · Zbl 1073.28015
[17] B. Riečan, D. Mundici, Probability on MV-algebras, in: Handbook of Measure Theory, vol. I, II, North-Holland, Amsterdam, 2002, pp. 869-909.
[18] Zadeh, L.A., Probability measures of fuzzy events, J. math. anal. appl., 23, 421-427, (1968) · Zbl 0174.49002
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