Issue:On the Poincaré recurrence theorem on IF-sets

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Title of paper: On two formulations of the IF state representation theorem Author(s): Jaroslav Považan Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia jaroslav.povazan@umb.sk Presented at: 11th International Workshop on Intuitionistic Fuzzy Sets, Banská Bystrica, Slovakia, 30 Oct. 2015 Published in: "Notes on Intuitionistic Fuzzy Sets", Volume 21, 2015, Number 5, pages 16–19 Download:  PDF (140  Kb, File info) Abstract: The Recurrence theorem by Poincaré is one of basic results of the standard ergodic theory. In classical sense the main structure is a σ-algebra of sets and the measure-preserving maps are represented by preimages of classical maps. In this article we change the σ-algebra by a family ℱ of Intuitionistic Fuzzy Sets (IF-sets), which were introduced by Krassimir T. Atanassov, and the probability by an IF-state. Keywords: recurrence theorem, IF-sets, s-preserving mappings, IF-state. AMS Classification: 03E72. References: Atanassov, K.T. (1999) Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Heidelberg. Bennett, M. K. & D. J. Foulis (1994) Effect algebras and unsharp quantum logics. Foundations of Physics, 24, 1331–1352. Dvurečenskij, A. (1976) On some properties of transformation of a logic. Math. Slovaca, 26, 131–137. Chovanec, F. & F. Kôpka (1994) D-posets. Math. Slovaca, 44, 21–34. Kôpka, F. (2008) Quasiproduct on Boolean D-posets. Int. J. Theor. Physics, 47, 26–35. Maličký, P. (2007) Category version of the Poincaré recurrence theorem. Topology Appl., 154, 2709–2713. Mundici, D. (2011) Advanced Lukasiewicz calculus and MV-algebras. Springer, New York. Poincaré, H. (1889) Les methodes nouvelles de la mechanique classique celeste, Vol. 3, Gauthiers-Villars, Paris. Riečan, B. (2013) Variation on a Poincaré theorem. Fuzzy Sets and Systems, 232, 39–45. Riečan, B. & D. Mundici (2002) Probability on MV-algebras. In: Handbook of Measure Theory (E. Pap. ed.), Elsevier Science, Amsterdam, 869– 909. Riečan, B. & T. Nebrunn (1997) Integral, Measure, and Ordering. Kluwer, Dordrecht. Citations: The list of publications, citing this article may be empty or incomplete. If you can provide relevant data, please, write on the talk page.