Fuzzy Ideals of Near-rings Based on the Theory of Falling Shadows

In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [4] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [13] introduced the theory of falling shadows which directly relates probability concepts with the membership function of fuzzy sets. Falling shadow representation theory shows us the way of selection related on the joint degrees distributions. It is reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. The mathematical structure of the theory of falling shadows is formulated. Tan et al. [11, 12] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. A near-ring satisfying all axioms of an associative ring, expect for commutativity of addition and one of the two distributive laws. Abou-Zaid [1] introduced the concepts of fuzzy subnear-rings (ideals) and studied some of their related properties in near-rings. Further, some properties were discussed by Hong and Kim et al. in [5] and [6], respectively. In [3], Davvaz introduced the concepts of (∈∈ ∨q)-fuzzy subnear-rings(ideals) of near-rings. Further, Zhan et al. [14, 15, 16, 17] investigated some generalized fuzzy ideals of near-rings. The other important results can be found in [2, 10]. Fuzzy near-rings are in particular designed for situations in which natural-language expressions need to be modelled without artificial specifications about borderline cases. Numerous applications where fuzzy near-rings have been