Abstract
In this paper, the cut sets, decomposition theorems and representation theorems of intuitionistic fuzzy sets and interval valued fuzzy sets are researched indail. First, new definitions of four kinds of cut sets on intuitionistic fuzzy sets are introduced, which are generalizations of cut sets on Zadeh fuzzy sets and have the same properties as that of Zadeh fuzzy sets. Second, based on these new cut sets, the decomposition theorems and representation theorems on intuitionistic fuzzy sets are established. Each kind of cut sets corresponds to two kinds of decomposition theorems and representation theorems. Thus eight kinds of decomposition theorems and representation theorems on intuitionistic fuzzy sets are obtained, respectively. At last, new definitions of cut sets on interval valued fuzzy sets are given based on the theory of cut sets on intuitionistic fuzzy sets, and eight kinds of decomposition theorems and representation theorems on interval valued fuzzy sets are also obtained. These results provide a fundamental theory for the research of intuitionistic fuzzy sets and interval valued fuzzy sets.
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References
Zadeh L A. Fuzzy sets. Inf Control, 1965, 8: 338–353
Wang GJ. L-fuzzy Topology Space Theory (in Chinese). Xi’an: Shanxi Normal University Press, 1988. 1–406
Liu Y M, Luo M K. Fuzzy Topology. Singapore: World Scientific Publishing, 1990. 1–306
Mordeson J N, Malik D S. Fuzzy Commutative Algebra. Singapore: World Scientific Publishing, 1998. 1–258
Mordeson J N, Bhutani K R, Rosenfeld A. Fuzzy Group Theory. New York: Springer, 2005. 1–292
Zhang G Q. Fuzzy Measure Theory (in Chinese). Guiyang: Guizhou Science and Technology Press, 1994. 1–265
Wu C X, Ma M. The Basis of Fuzzy Analysis (in Chinese). Beijing: National Defence Industry Press, 1991. 1–147
Bertoluzza C, Solci M, Capodieci M L. Measure of a fuzzy set: the α-cut approach in the finite case. Fuzzy Sets Syst, 2001, 123: 93–102
Garcia J N, Kutalik Z, Cho K H, et al. Level sets and minimum volume sets of probability density functions. Int J Appr Reason, 2003, 34: 25–47
Pap E, Surla D. Lebesgue measure of α-cuts approach for finding the height of the membership function. Fuzzy Sets Syst, 2000, 111: 341–350
Lai Y J, Hwang C L. Fuzzy Mathematical Programming-Methods and Applications. Berlin: Springer-Verlag, 1992. 1–156
Xu Z S. Uncertain Multiple Attribute Decision Making: Methods and Applications (in Chinese). Beijing: Tsinghua University Press, 2004. 3–236
Dubois D, Hüllermeier E, Prade H. On the representation of fuzzy rules in terms of crisp rules. Inf Sci, 2003, 151: 301–326
Luo C Z, Wang P Z. Representation of compositional relations in fuzzy reasoning. Fuzzy Sets Syst, 1990, 36: 77–81
Wang G J. Non-classical Logic and Approximate Reasoning (in Chinese). Beijing: Science Press, 2000. 24–185
Luo C Z. Introduction to Fuzzy Sets (1) (in Chinese). Beijing: Beijing Normal University Press, 1989. 1–486
Yuan X H, Li H X, Luo C Z. New cut sets and their Applications (in Chinese). Fuzzy Syst Math, 1997, 24: 37–43
Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst, 1986, 20: 87–96
Zadeh L A. Outline of a new approach to the analysis of complex systems and decisi on processes, interval-valued fuzzy sets. IEEE Trans Syst Man Cybernet, 1973, 3: 28–44
Wang G J, He Y Y. Intuitionistic fuzzy sets and L-fuzzy sets. Fuzzy Sets Syst, 2000, 110: 271–274
Yang W C, Yin M E. Relations between some L-fuzzy sets (in Chinese). J Liaoning Norm Univ (Nat Sci Ed), 2005, 28: 143–154
Li M. Cut sets of intuitionistic fuzzy sets (in Chinese). J Liaoning Norm Univ (Nat Sci Ed), 2007, 30: 152–154
Zeng W Y, Li H X, Shi Y. Decomposition theorem of interval value fuzzy sets (in Chinese). J Beijing Norm Univ (Nat Sci Ed), 2003, 39: 171–177
Zeng W Y, Li H X, Shi Y. Representation theorem of interval value fuzzy sets (in Chinese). J Beijing Norm Univ (Nat Sci Ed), 2003, 39: 444–447
Zhao Y B, Zeng W Y, Li H X. Extension theorem of interval value fuzzy sets (in Chinese). J Beijing Norm Univ (Nat Sci Ed), 2007, 43: 1–5
Luo C Z. Introduction to Fuzzy Sets (2)(in Chinese). Beijing: Beijing Normal University Press, 1991. 1–291
Yuan X H, Wu Z H. Axiomatic description of cut sets of L-fuzzy set (in Chinese). J Liaoning Norm Univ (Nat Sci Ed), 1999, 22: 4–9
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Yuan, X., Li, H. & Sun, K. The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets. Sci. China Inf. Sci. 54, 91–110 (2011). https://doi.org/10.1007/s11432-010-4078-6
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DOI: https://doi.org/10.1007/s11432-010-4078-6