Abstract
In this paper, we established decomposition theorems and representation theorems of interval-valued intuitionistic fuzzy sets(IVIFS) by use of cut sets of interval-valued intuitionistic fuzzy sets. We have shown that 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 interval-valued intuitionistic fuzzy sets are obtained, respectively. These results provide a fundamental theory for the research of interval-valued intuitionistic fuzzy sets.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 352–378 (1965)
Wang, G.J.: L-fuzzy Topology Space Theory. Shanxi Normal University Press, Xian (1988) (in Chinese)
Liu, Y.M., Luo, M.K.: Fuzzy Topology. World Scientific Publishing, Singapore (1990)
Mordeson, J.N., Malik, D.S.: Fuzzy Commutative Algebra. World Scientific Publishing, Singapore (1998)
Mordeson, J.N., Bhutani, K.R., Rosenfeld, A.: Fuzzy Group Theory. Springer, New York (2005)
Zhang, G.Q.: Fuzzy Measure Theory. Guizhou Science and Technology Press, Guiyang (1994) (in Chinese)
Wu, C.X., Ma, M.: The Basis of Fuzzy Analysis. National Defence Industry Press, Beijing (1991) (in Chinese)
Bertoluzza, C., Solci, M., Capodieci, M.L.: Measure of a fuzzy set: the α-cut approach in the finite case. Fuzzy Sets and Systems 123(1), 93–102 (2001)
Garcia, J.N., Kutalik, Z., Cho, K.H., et al.: Level sets and minimum volume sets of probability density functions. International Journal of Approximate Reasoning 34(1), 25–47 (2003)
Pap, E., Surla, D.: Lebesgue measure of α-cuts approach for finding the height of the membership function. Fuzzy Sets and Systems 111(3), 341–350 (2000)
Lai, Y.J., Hwang, C.L.: Fuzzy Mathematical Programming-Methods and Applications. Springer, Berlin (1992)
Xu, Z.S.: Uncertain Multiple Attribute Decision Making: Methods and Applications. Tsinghua University Press, Beijing (2004) (in Chinese)
Dubois, D., Hüllermeier, E., Prade, H.: On the representation of fuzzy rules in terms of crisp rules. Information Sciences 151, 301–326 (2003)
Luo, C.Z., Wang, Z.P.: Representation of compositional relations in fuzzy reasoning. Fuzzy Sets and Systems 36(1), 77–81 (1990)
Wang, G.J.: Non-classical Logic and Approximate Reasoning. Science Press, Beijing (2000) (in Chinese)
Luo, C.Z.: Introduction to Fuzzy Sets (1). Beijing Normal University Press (1989)
Yuan, X.H., Li, H.X., Lee, E.S.: Three new cut sets of fuzzy sets and new theories of fuzzy sets. Computers and Mathematics with Applications 57, 691–701 (2009)
Yuan, X.H., Li, H.X., Sun, K.B.: Theory based on interval-valued level cut sets of Zadeh fuzzy sets. Advances in Intelligent and Soft Computing, vol. 62, pp. 501–510. Springer, Heidelberg (2009)
Goguen, J.A.: L-fuzzy Sets. J. Math. Anal. Appl. 18, 145–174 (1967)
Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decesion processes interval-valued fuzzy sets. IEEE Trans. Syst. Man Cybernet. 3, 28–44 (1973)
Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, 87–96 (1986)
Atanassov, K.: Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems 31, 343–349 (1989)
Li, X.S., Yuan, X.H., Lee, E.S.: The three-dimensional fuzzy sets and their cut sets. Computers and Mathematics with Applications 58, 1349–1359 (2009)
Shang, Y.G., Yuan, X.H., Lee, E.S.: The n-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets. Computer and Mathematics with Applications (2010), doi:10,1016/j.camwa,2010,04,044
Mendel, J.M.: Advances in type-2 fuzzy sets and systems. Information Sciences 177, 84–110 (2007)
Yuan, X.H., Li, H.X., Sun, K.B.: The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval-valued fuzzy Sets. Science in China, Series F 39(9), 933–945 (2009) (in Chinese)
Yuan, X.H., Li, H.X.: Cut sets on interval-valued intuitionistic fuzzy sets. In: Sixth International Conference On Fuzzy Systems and Knowledge Disvovery, vol. 6, pp. 167–171. IEEE Computer Society, Los Alamitos (2009)
Yuan, X.H., Li, H.X., Lee, E.S.: On the definition of intuitionistic subgroups. Computers and Mathematics with Applications 59(9), 3117–3129 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yuan, XH., Li, HX. (2010). Decomposition Theorems and Representation Theorems on the IVIFS. In: Cao, By., Wang, Gj., Chen, Sl., Guo, Sz. (eds) Quantitative Logic and Soft Computing 2010. Advances in Intelligent and Soft Computing, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15660-1_32
Download citation
DOI: https://doi.org/10.1007/978-3-642-15660-1_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15659-5
Online ISBN: 978-3-642-15660-1
eBook Packages: EngineeringEngineering (R0)