Pythagorean fuzzy sets are an extension of the intuitionistic fuzzy sets, and they also overcome the limitations of the intuitionistic fuzzy sets. The theory of Pythagorean fuzzy sets possesses significant advantages in handling vagueness and complex uncertainty. Additionally, Pythagorean fuzzy information is used to simulate the ambiguous nature of subjective judgements and measure the fuzziness and impression more flexibly. This chapter presents some useful concepts of Pythagorean fuzzy sub-ring, upper, and lower cut seta. Also, the authors derive some useful properties of Pythagorean Q-fuzzy ideals of near-rings.
Top1 Introduction
Considering the imprecision in decision-making, Zadeh (1965) introduced the idea of fuzzy set which has a membership function, 𝜇 that assigns to each element of the universe of discourse, a number from the unit interval [0,1] to indicate the degree of belongingness to the set under consideration. The notion of fuzzy sets generalizes classical sets theory by allowing intermediate situations between the whole and nothing. In a fuzzy set, a membership function is defined to describe the degree of membership of an element to a class. The membership value ranges from 0 to 1, where 0 shows that the element does not belong to a class, 1 means belongs, and other values indicate the degree of membership to a class. For fuzzy sets, the membership function replaced the characteristic function in crisp sets. Nasseri et. al. (2015) had done a lot research work in the field of fuzzy set theory and they also used some interesting method for ranking triangular fuzzy numbers and trapezoidal fuzzy number.
Albeit, the concept of fuzzy sets theory seems to be inconclusive because of the exclusion of nonmembership function and the disregard for the possibility of hesitation margin. The theory of Instuitionistic fuzzy set has been found to be more useful to deal with vagueness and uncertainty in decision situations than that of the fuzzy set. Atanassov (1986) critically studied these shortcomings and proposed a concept called intuitionistic fuzzy sets (IFSs). The construct (that is, IFSs) incorporates both membership function, 𝜇 and nonmembership function, 𝜈 with hesitation margin, 𝜋 (that is, neither membership nor nonmembership functions), such that 𝜇+𝜈≤1 and 𝜇+𝜈+𝜋=1. The notion of IFSs provides a flexible framework to elaborate uncertainty and vagueness. The idea of IFS seems to be resourceful in modeling many real-life situations like medical diagnosis, career determination, selection process, and multi-criteria decision-making, among others. Adak et.al., (2011, 2012, 2013, 2014) have done of research work in the field IFSs.
There are situations where 𝜇+𝜈≥1 unlike the cases capture in IFSs. This limitation in IFS naturally led to a construct, called Pythagorean fuzzy sets (PFSs). Pythagorean fuzzy set (PFS) proposed a new tool to deal with vagueness considering the membership grade, 𝜇 and nonmembership grade, 𝜈 satisfying the conditions 𝜇+𝜈≤1 or 𝜇+𝜈≥1, and also, it follows that 𝜇2+𝜈2+𝜋2=1, where 𝜋 is the Pythagorean fuzzy set index. The construct of PFSs can be used to characterize uncertain information more sufficiently and accurately than IFS. Garg (2016) presented an improved score function for the ranking order of interval-valued Pythagorean fuzzy sets (IVPFSs). Based on it, a Pythagorean fuzzy technique for order of preference by similarity to ideal solution (TOPSIS) method by taking the preferences of the experts in the form of interval-valued Pythagorean fuzzy decision matrices was discussed.
Pythagorean fuzzy set has attracted great attentions of many researchers, and subsequently, the concept has been applied to many application areas such as decision-making, aggregation operators, and information measures.
Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences and so on. This provides sufficient motivations to researchers to review various concepts and results from the real abstract algebra in the broader framework of fuzzy setting.
R.Biswas and S.Nanda (1994) introduced the concept of lower and upper approximation of a subgroup of a group. N.Kuroki (1997) gave the notion of rough ideal in a semigroup. B.Davvaz (2006, 2008) has introduced roughness in rings and the notion of rough prime ideals and rough primary ideals in rings. The authors V.Selvan and G.Senthil Kumar (2012) have studied the lower and upper approximations of ideals and fuzzy ideals in a semiring. The notion of rough intuitionistic fuzzy ideal in a semi group was given by Gosh et. al., (2012).