The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.

[1]  John N. Mordeson,et al.  Fuzzy commutative algebra , 1998 .

[2]  Hans-Jürgen Zimmermann,et al.  Introduction to Fuzzy Sets , 1985 .

[3]  M..  Introduction to Fuzzy Sets 1 n Pavement Evaluation : , .

[4]  Azriel Rosenfeld,et al.  Fuzzy Group Theory , 2005, Studies in Fuzziness and Soft Computing.

[5]  Carlo Bertoluzza,et al.  Measure of a fuzzy set. The [alpha]-cut approach in the finite case , 2001, Fuzzy Sets Syst..

[6]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[7]  E. Stanley Lee,et al.  Three new cut sets of fuzzy sets and new theories of fuzzy sets , 2009, Comput. Math. Appl..

[8]  Hongxing Li,et al.  The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets , 2010, Science China Information Sciences.

[9]  C.-Z. Luo,et al.  Representation of compositional relations in fuzzy reasoning , 1990 .

[10]  Wolfgang Näther,et al.  On possibilistic inference , 1990 .

[11]  H. Carter.  Fuzzy Sets and Systems — Theory and Applications , 1982 .

[12]  Lotfi A. Zadeh,et al.  Outline of a New Approach to the Analysis of Complex Systems and Decision Processes , 1973, IEEE Trans. Syst. Man Cybern..

[13]  Eyke Hüllermeier,et al.  On the representation of fuzzy rules in terms of crisp rules , 2003, Inf. Sci..

[14]  Guo-Jun Wang,et al.  Intuitionistic fuzzy sets and L-fuzzy sets , 2000, Fuzzy Sets Syst..

[15]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[16]  Kwang-Hyun Cho,et al.  Level sets and minimum volume sets of probability density functions , 2003, Int. J. Approx. Reason..

[17]  J. Goguen.  L-fuzzy sets , 1967 .

[18]  Dusan Surla,et al.  Lebesgue measure of α-cuts approach for finding the height of the membership function , 2000, Fuzzy Sets Syst..