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Department of Mathematics, Chikkanna Govt. Arts College, Tiruppur
This paper presents the concepts of the Fermatean m-Polar Fuzzy order of an element and the Fermatean m-Polar Fuzzy order of a Fermatean m-Polar Fuzzy Subgroup are introduced and investigated. In addition the algebraic properties of Fermatean m-Polar Fuzzy Cosets and Fermatean m-Polar Fuzzy Normal Subgroups (FmPFNSG) are studied, leading to several characterization results and structural theorems. Illustrative examples are included.
Zadeh [10] introduced the concept of fuzzy sets in 1965, establishing a mathematical framework for representing uncertainty through membership functions. This pioneering work laid the foundation for the subsequent development of fuzzy algebra and fuzzy group theory.
Biswas [4] extended Zadeh's fuzzy set theory to algebraic structures by defining fuzzy subgroups and anti-fuzzy subgroups. This work initiated the study of fuzzy groups and inspired further investigations into their structural properties.
Biswas and Choudhury [3] examined the fundamental concepts of fuzzy groups and fuzzy subgroups, presenting several algebraic properties and illustrating how classical group-theoretic notions can be generalized within the fuzzy environment.
Senapathi and Yager [8] introduced Fermatean fuzzy sets as a more flexible extension of intuitionistic and Pythagorean fuzzy sets. Their model allows a wider range of membership and non-membership values while preserving the Fermatean constraint, making it suitable for handling higher levels of uncertainty.
Silambarasan [9] applied the theory of Fermatean fuzzy sets to group theory by defining Fermatean fuzzy subgroups and establishing several of their basic algebraic properties. This work demonstrated the applicability of Fermatean fuzzy concepts in abstract algebra.
Naeem, Riaz, and Karaaslan [5] proposed the notion of Pythagorean m-polar fuzzy sets and investigated their theoretical properties. They also demonstrated the usefulness of these sets in solving decision-making problems involving multiple attributes.
Abdul Razaq [1] studied Pythagorean fuzzy normal subgroups and Pythagorean fuzzy isomorphisms in detail. The paper established important characterizations of normality and homomorphic mappings, thereby enriching the algebraic theory of Pythagorean fuzzy groups.
Bhunia [2] extended the classical Lagrange theorem to Pythagorean fuzzy subgroups by introducing an appropriate notion of subgroup order in the Pythagorean fuzzy setting. The results provide a significant connection between classical finite group theory and Pythagorean fuzzy algebra.
Radharamani and Rajeswari [6] investigated the fundamental properties of Fermatean m-polar fuzzy sets, including their operations and structural characteristics. The study provides a theoretical basis for extending Fermatean fuzzy concepts to more advanced algebraic systems.
Radharamani and Rajeswari [7] introduced Fermatean m-polar fuzzy subgroups and established several of their algebraic properties. Their work extends the theory of Fermatean m-polar fuzzy sets to group theory and provides a foundation for further research on concepts such as normal subgroups, cosets, and group order in the Fermatean m-polar fuzzy framework.
2.Preliminaries
Definition :2.1[10] Let U
be an universal set. A fuzzy set F
on U
is defined by F
=u,φu:for all u∈U ,where φ
is a function from U⟶0,1
and it is called the membership function.
Definition :2.2[3] Let (Ǵ, *)
be a group and U be an universal set. The fuzzy subset F
=u,φu
of a group Ǵ
is said to be a fuzzy subgroup of Ǵ
if it satisfies the following conditions
Definition :2.3[5] Let m
be any positive integer. A Pythagorean m-polar fuzzy set (PmFS) on U
is defined by
Definition :2.7[7] Let (Ǵ , *) be a group and F=φF(i)u,ΨF(i)u,i=1,2,…,m be a FmPFS. Then F is said to be a Fermatean m-Polar Fuzzy Subgroup (FmPFSG) of G if it satisfies the following conditions
Here φFi3u=φFiu3 and ΨFi3u=ΨFiu3
Definition :2.8[1] Let P=u,φPu,ΨPu:u∈Ǵ be a Pythagorean Fuzzy Subgroup of Ǵ (PFSG). For any element a∈Ǵ the Pythagorean fuzzy left coset of P associated with a is represented by aP=u,φaPu,ΨaPu:u∈Ǵ , where φaPu2=φPa-1u2 and ΨaPu2=ΨPa-1u2 . Likewise, the Pythagorean fuzzy right coset of P corresponding to the element a and is denoted by Pa=u,φPau,ΨPau:u∈Ǵ , where ΨPau2=ΨPua-12 and ΨPau2=ΨPua-12 .
Definition :2.9[1] Let PFSG P=u,φPu,ΨPu:u∈Ǵ of Ǵ .Then P is said to be a Pythagorean fuzzy normal subgroup of Ǵ if aP=Pa for all a∈Ǵ .
Theorem:2.10[2] Let PFSG P=u,φPu,ΨPu:u∈Ǵ of Ǵ and v∈Ǵ .Then the subset Γv=u∈Ǵ/ φP2(v)≥φP2u and ΨP2(v)≤ΨP2u is a Pythagorean fuzzy subgroup of Ǵ .
Definition:2.10[2] Let PFSG P=u,φPu,ΨPu:u∈Ǵ of Ǵ and v∈Ǵ .Then the subgroup Γv=u∈Ǵ/ φP2(v)≥φP2u and ΨP2(v)≤ΨP2u is called the Pythagorean fuzzy semi-level subgroup of Ǵ corresponding to v .
Definition:2.11[2] Suppose P=φPu,ΨPu is a PFSG of Ǵ and a∈Ǵ .Then the Pythagorean fuzzy order (PFO) of a in P is represented by PFOaP , is taken to be the order of the corresponding semi-level subgroup. That is PFOaP=oΓa for all a∈Ǵ .
3.Fermatean m-Polar Fuzzy Cosets
Definition :3.1
Let Ǵ,* be a group and F=φFiu,ΨFiu,i=1,2,3,…,m be a FmPFSG. Let a∈Ǵ , then the Fermatean m-Polar fuzzy left coset of F is defined by
aF=u,φaFiu,ΨaFiu,,i=1,2,…,m , for all u∈Ǵ ,
where φaFiu3=φFia-1u3 and ΨaFiu3=ΨFia-1u, 3 for all u∈G and also the Fermatean m-Polar fuzzy right coset of F is defined by
Fa={φFaiu,ΨFaiu,i=1,23,…,m,} , for all u∈Ǵ ,
where φFaiu=φFiua-1 and ΨFaiu=ΨFiua-1 , for all u∈Ǵ.
Definition :3.2
Let Ǵ,* be a group and F=φFiu,ΨFiu,i=1,2,3,…,m be a FmPFSG of Ǵ,*.
Then F is said to be Fermatean m-Polar fuzzy normal subgroup Ǵ (FmPFNSG) if aF=Fa for all u∈Ǵ .
Theorem:3.3
Assume F=φFiu,ΨFiu,i=1,2,3,…,m be a FmPFSG on Ǵ,* and u,v∈Ǵ. Then
Ὠu=v∈Ǵ / φFi3v≥φFi3u and ΨFi3v≤ΨFi3u,i=1,2,3,…,m
is a subgroup of Ǵ,* .
Proof:
Let F=φFiu,ΨFiu,i=1,2,3,…,m be a FmPFSG on Ǵ,* and
Ὠu=v∈Ǵ / φFi3v≥φFi3u and ΨFi3v≤ΨFi3u,i=1,2,3,…,m
where u,v∈Ǵ.
Since φFi3e≥φFi3u and ΨFi3e≤ΨFi3u , where 'e' is the identity element of Ǵ,*, then e∈Ὠu and also Ὠu⊆ Ǵ . Let u,v∈Ǵ.
φFi3u*v-1≥φFi3u⋀φFi3v≥φFi3u and
ΨFi3u*v-1≤ΨFi3u⋁ΨFi3v≤ΨFi3u
This implies, u*v-1∈Ὠu
Therefore, Ὠu is a subgroup of Ǵ,*.
Definition :3.4
Let F=φFiu,ΨFiu,i=1,2,3,…,m be a FmPFSG on Ǵ,* .Then Ὠu is called the Fermatean m-polar semi level subgroup of Ǵ,* corresponding to u .
Definition:3.5
Let F=φFiu,ΨFiu,i=1,2,3,…,m be a FmPFSG on Ǵ,* and a∈Ǵ. Then the Fermatean m-polar fuzzy order of a is defined by the number of elements in the Fermatean m -polar semi level subgroup of Ǵ,* corresponding to a and it is denoted by FmPFO(a) .
i.e.,FmPFOa=O(Ὠa) , for all a∈Ǵ.
Example:3.6 Let us consider the set Ǵ=Z4=0,1,2,3 ,then Ǵ ,⊕4 is a group, where ‘⊕4 ' is addition modulo 4. Now we define a F3PFS F=u,φF(i)u,ΨF(i)u,i=1,2 ,3
on Ǵ by
F=0,1,0,0.8,0.30.7,0.4,1,0.5,0.4,0.4,0.50.2,0.6,2,1,0,0.8,0.30.7,0.4,3,0.5,0.4,(0.4,0.5)(0.2,0.6) .
Clearly, F is a Fermatean m-Polar Fuzzy Subgroup on Ǵ ,⊕4 .Then FmPFO of the elements of Z4 in F is presented by
FmPFO(0)=OὨ0=2 , FmPFO(1)=OὨ1=4
FmPFO(2)=OὨ2=2 , FmPFO(3)=OὨ3=4
From above example, we see that FmPFO0≠O0 .
Remark.3.7 The FmPFO of an element in FmPFSG on a group Ǵ may not always be same to the element’s order in the group.
Definition:3.8 Let F be a FmPFSG on a group Ǵ,*. The index of F is the number of distinct left coset of F in Ǵ and is denoted by Ǵ :F
REFERENCES
A. Radharamani, S. Rajeswari*, A Comprehensive Study On Fermatean m-Polar Fuzzy Cosets, Fermatean m-Polar Fuzzy Normal Subgroups, Int. J. Sci. R. Tech., 2026, 3 (8), 366-378. https://doi.org/10.5281/zenodo.21873442
10.5281/zenodo.21873442