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On invariant \(IF\)-state. (English) Zbl 1398.28009

Summary: The Haar measure on invariant state for fuzzy sets is constructed in a locally compact space. Moreover, the invariant state is studied on \(MV\)-algebra generated by a family of intuitionistic fuzzy sets, important as well as from the theoretical point of view as from the applications.

MSC:

28E10 Fuzzy measure theory
06D35 MV-algebras
22D05 General properties and structure of locally compact groups
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References:

[1] Atanassov K (1999) Intuitionistic fuzzy sets. Physica, Heidelberg, pp 1-137 · Zbl 1170.03337
[2] Ciungu L, Riečan B (2009) General form of probabilities on IF-sets, fuzzy logic and applications. In: Proceedings of the WILF Palermo, pp 101-107 · Zbl 1246.60006
[3] Ciungu, L; Riečan, B, Representation theorem for probabilities on IFS-events, Inf Sci, 180, 793-798, (2010) · Zbl 1196.60012 · doi:10.1016/j.ins.2009.11.003
[4] Čunderlíková K (2008) The individual ergodic theorem on IF events. Soft computing a fusion of foundations, methodologies and applications, generalized nets and related topics, vol I, 3360. EXIT, Warsaw
[5] De, SK; Biswas, R; Roy, AR, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets Syst, 117, 209-213, (2001) · Zbl 0980.92013 · doi:10.1016/S0165-0114(98)00235-8
[6] \(\check{\text{D}}\)urica M (2010) Hudetz entropy on IF events. Developments in fuzzy sets, intuitionistic fuzzy sets, generalized nets, and related topics, vol I. IBS PAN SRI PAS, Warsaw, pp 73-86
[7] Halmos PR (1950) Measure theory. Van Nostrand, New York · Zbl 0040.16802 · doi:10.1007/978-1-4684-9440-2
[8] Jurečková, M; Riečan, B, On invariant observables and the individual ergodic theorem, Int J Theor Phys, 4, 1587-1597, (2005) · Zbl 1119.37006
[9] Lašová L (2010) The individual ergodic theorem on IF events. Developments in fuzzy sets, intuitionistic fuzzy sets, generalized nets, and related topics, vol I. IBS PAN SRI PAS, Warsaw, pp 131-140
[10] Markechová D, Riečan B, Rényi entropy and Rényi divergence in the intuitionistic fuzzy case. Entropy (to appear) · Zbl 1505.28020
[11] Markechová, D; Stehlíková, D; Tirpáková, A, The uniqueness of a left Haar measure in topological IP-loops, Mathematica Slovaca, 66, 1-8, (2016) · Zbl 1413.28022 · doi:10.1515/ms-2016-0205
[12] Michalíková A Riečan B (2017) On the Lebesgue IF measure, notes on intuitionistic fuzzy sets. ISSN: 1310-4926, pp 8-12 · Zbl 1398.28008
[13] Riečan, B, On a problem of radko mesiar: general form of IF-probabilities, Fuzzy Sets Syst, 152, 1485-1490, (2006) · Zbl 1098.60001
[14] Riečan B (2009) Poincaré recurrence theorem in MV-algebras. In: Prov. IFSA/EUSFLAT congress Lisabon, pp 1880-1881
[15] Riečan B (2012) In: Koleshko V (ed) Analysis of fuzzy logic models, intelligent systems. INTECH, London, pp 217-244
[16] Riečan, B, Variations on a Poincaré theorem, Fuzzy Sets Syst, 232, 39-45, (2013) · Zbl 1314.60031 · doi:10.1016/j.fss.2013.07.024
[17] Schwarz, Š, On the existence of invariant measures on certain type of bicompact semigroups (in Russian), Czech Math J, 7, 165-182, (1957) · Zbl 0089.01203
[18] Stehlíková, B; Markechová, D; Tirpáková, A, On the existence of a Haar measure in topological IP-loops, Kybernetika, 47, 740-754, (2011) · Zbl 1242.28022
[19] Szmidt, E; Kacprzyk, J, Intuitionistic fuzzy sets in some medical applications, Notes IFS, 7, 58-64, (2001) · Zbl 1087.92501
[20] Zadeh, L, Fuzzy sets, Inf Control, 8, 338-358, (1965) · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
[21] Zadeh, L, Probability measures of fuzzy events, J Math Anal Appl, 23, 421-427, (1968) · Zbl 0174.49002 · doi:10.1016/0022-247X(68)90078-4
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