Q-Fuzzification Of Fermatean Fuzzy Points And Local Properties Of Uncertainty Subgroup

The concept of fuzzy set is first introduced by L.A.Zadeh in 1965 [24]. After the tremendous development intuitionistic fuzzy set [4], pythagoren fuzzy set [23], (3,2) fuzzy set[15], picture fuzzy set [14], Neutrosophoic fuzzy set [19] was studied. Recently fermatean fuzzy  set [19] was studied by Yager and its application is briefly explained by [23]. Due to the inadequacy of FST in the sense that it admits only the membership grade, intuitionistic fuzzy sets (IFSs) was introduced [5, 6] by incorporating membership and non-membership grades with the chance for hesitation margin. Various properties of IFSs were presented  [4,5] and IFSs have been applied in real-life problems [4, 5, 6]. Consequently, Biswas [8, 9] introduced intuitionistic fuzzy subgroups (IFSGs) based on IFSs, Ahn et al. [2] deliberated on sublattice of lattice of IFSGs of a group, some properties of IFSGs were discoursed in [1]. In addition, Yuan et al. [22] shared some light on the description of IFSGs, Bal et al. [7] presented a brief note of kernel subgroups on IFGs and derived some properties of IFGs. Senapati.T [19] introduced fermatean fuzzy sets in 2020, an extension of intuitionistic fuzzy set (IFSs) and pythagorean fuzzy set (PFSs). AVs and NAVs of fermatean fuzzy sets reveal their dependence on greater powers with sum of cubes is less than 1. Ibrahim.H.Z [15] introduced and applied n, m-rung orthopair fuzzy sets in MCDM.. The concepts of fuzzy modules and fuzzy sub-modules was introduced by Negoita and Ralescu [16] in 1975. The concept of essential fuzzy modules was introduced by Hadi [13] in 2000. Using this idea. Abbas established the concept of essential fuzzy sub-modules and uniform fuzzy modules in 2012 Nagarajan and Solairaju and   [17], new Structure and Constructions of Q-fuzzy groups In this paper, we studied the notion of a fermatean Q-fuzzy point to study for the first time the notion of fermatean  Q-fuzzy subgroups and its local properties.  This new concept showed us to reformulate all the mathematical properties of fermatean Q-fuzzy subgroups structure.

2. Preliminaries

In this section, we will present some notions and results to use them in the  sequal.

Definition-2.1: [24] Let X  be an universe of discourse. Then a fuzzy set A  is an object having the following formulation: A=x, μAx/x∈X  where μA:X→0, 1  and μAx  is called the member degree of x  in X .

Example 2.2: Let X=a, b,c.  Then the fuzzy set is defined as

Definition-2.3: [17] A Q-fuzzy set is defined as a mapping μ:X×Q→0, 1 , assigning a degree of membership between 0 and 1 to each element x∈X  for each parameter q∈Q .

Definition-2.4:[17] Let X  be a non-empty set. An intuitionistic Q-fuzzy set (IQFS for short) of X  defined as an object having the form A=x,Jx,q,Kx,q/x∈X,q∈Q , where

J:X×Q→0, 1  and K:X×Q→0, 1  denote the degree of membership (namely Jx,q ) and the degree of non-membership (namely Kx,q ) of each element x∈X  it to the set A , respectively and 0≤Jx,q+Kx,q≤1  for each x∈X  and q∈Q . For the sake of simplicity we shall use the symbol A=J,K  for the intuitionistic Q-fuzzy set

A=x,Jx,q,Kx,q/x∈X,q∈Q .

We shall point out the membership Grades (MG’s) related Fermatean fuzzy sets as Fermatean membership grades.

Theorem 2.8: [19] The set of FMG’s is greater than the set of pythagorean membership grades (PMG’s) and intuitionistic membership grades (IMG’s).

Proof:  This development can be evidently recognized in the following figure.

CONCLUSION

In this paper, we have presented a new definition of fermatean Q-fuzzy subgroups(FQFSG) using the concept of fermatean Q-fuzzy points and with the help of these basic concepts. We have exposed to show some local properties on fermatean Q-fuzzy subgroups.

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