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Cut Set Theorems for Rectangular 𝑳-fuzzy Complex Numbers and 𝑳-multi-fuzzy Complex Numbers
Pishtiwan O. Sabir1*&Aram N. Qadir2 

1 Department of Mathematics, College of Science, University of Sulaimani, Sulaimani, Iraq 
2 Department of Mathematics, College of Education, University of Garmian, Kalar, Iraq 

*Corresponding author Email: 

In this paper, the properties of the lower cut set and the upper cut set of rectangular 𝐿-fuzzy complex numbers are studied and it is showed that the (ℓ)- upper cut and [ℓ]-upper cut on this type of fuzzy numbers are satisfied under 𝛽 and 𝛼 preserving mapping, respectively. The inclusion properties on the set of cut set types are proved and some decompositions of this type of number and its characterizations are discussed. The basic arithmetic operations between rectangular 𝐿-fuzzy complex numbers are introduced and some representations of them are obtained. Furthermore, the notions of rectangular 𝐿-multi-fuzzy complex numbers are demonstrated and some fundamental theorems and rules were presented for calculating binary operations between them. Some modulus and inequalities on the set of 𝐿-multi-fuzzy quantities are derived.

Key Words: Cut set, 𝐿-fuzzy set, 𝐿- fuzzy number, 𝐿-fuzzy complex number, 𝐿- multi-fuzzy complex number. 

[1] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965), no. 3, 338–353. [2] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967), no. 1, 145–174. [3] H.L. Huang and F.G. Shi, 𝐿-fuzzy numbers and their properties, Information Sciences 178 (2008), no. 4, 1141–1151 [4] F.G. Shi, Theory of 𝐿𝛼-nest sets and 𝐿𝛽-nest sets and their applications, Fuzzy Syst. Math. 4 (1995) 65–72. [5] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), no. 1, 87–96. [6] R.R. Yager, On the theory of bags, Int. J. Gen. Syst., 13 (1986), no. 1, 23–37. [7] S. Miyamoto, Fuzzy multisets and their generalizations. In Multiset Processing; Springer: Berlin, Germany (2001) 225–235. [8] P.O. Sabir, Notes on extension of fuzzy complex sets, Tikrit J. Pure Sci., 22 (2017), no. 9, 88-93. [9] S.S.L. Chang and L.A. Zadeh, On fuzzy mappings and control, IEEE Trans. Syst., Man and Cybern., SMC.2, 1 (1972), 30-34. [10] A. Dey and M. Pal, Multi-fuzzy complex numbers and multi-fuzzy complex sets, Int. J. Fuzzy Syst. Appl. 4 (2015), no. 2, 15-27. [11] J.J. Buckley and E. Eslami, An introduction to fuzzy logic and fuzzy sets, Physica-Verlag Heidelberg, New York, 2002. [12] G. Birkhoff, Lattice theory, Amer. Math. Soc., Providence, R.I., 3rd ed., 1967. [13] G. Gierz, et al., A compendium of continuous lattices, Springer-Verlag, Berlin, 1980. [14] G.J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems 47 (1992), no. 3, 351–376. [15] G.Q. Wu, P. Du, and F.G. Shi, Decomposition theorems and representation theorem of L-fuzzy field, J. Fuzzy Math. 3 (1997) 353–359. [16] F.G. Shi, L-metric on the space of L-fuzzy numbers, Fuzzy Sets and Systems 399 (2020) 95–109. [17] W.D. Blizard, Multiset theory, Notre Dame J. Form. Log. 30 (1989), no. 1, 36-66. [18] S. Sebastian and T.V. Ramakrishnan, Multi-fuzzy sets: An extension of fuzzy sets, Fuzzy inf. Eng. 3 (2011), no. 1, 35-43