In this paper, we initiate the novel concept of complex intuitionistic fuzzy subgroups and prove that every complex intuitionistic fuzzy subgroup generates two intuitionistic fuzzy subgroups. We extend this ideology to define the concept of level subsets of complex intuitionistic fuzzy set and discuss its various fundamental algebraic characteristics. We also show that the level subset of the complex intuitionistic fuzzy subgroups is a subgroup. Furthermore, we investigate the homomorphic image and preimage of complex intuitionistic fuzzy subgroup under group homomorphism. Moreover, we prove that the product of two complex intuitionistic fuzzy subgroups is also a complex intuitionistic fuzzy subgroup and develop some new results about direct product of complex intuitionistic fuzzy subgroups.INDEX TERMS Complex intuitionistic fuzzy set, Complex intuitionistic fuzzy subgroup, Level subsets, Product of complex intuitionistic fuzzy sets, Product of complex intuitionistic fuzzy subgroups.
In this study, the t-intuitionistic fuzzy normalizer and centralizer of t intuitionistic fuzzy subgroup are proposed. The t-intuitionistic fuzzy centralizer is normal subgroup of t-intuitionistic fuzzy normalizer and investigate various algebraic properties of this phenomena. We also introduce the concept of t-intuitionistic fuzzy Abelian and cyclic subgroups and prove that every t-intuitionistic fuzzy subgroup of Abelian (cyclic) group is t-intuitionistic fuzzy Abelian (cyclic) subgroup. We show that the image and preimage of t-intuitionistic fuzzy Abelian (cyclic) subgroup are t-intuitionistic fuzzy Abelian (cyclic) subgroup under group homomorphism. INDEX TERMS t-intuitionistic fuzzy set, t-intuitionistic fuzzy subgroup, t-intuitionistic fuzzy Abelian subgroup, t-intuitionistic fuzzy cyclic subgroup AMS(MOS) Subject Classifications: 03F55, 08A72, 20N25
In this paper, we introduce idea of complex fuzzy subfield and discuss its various algebraic aspects. We prove that every complex fuzzy subfield generate two fuzzy fields and shows that intersection of two complex fuzzy subfields is also complex fuzzy subfields. We also present the concept of level subsets of complex fuzzy subfield and shows that level subset of complex fuzzy subfield form subfield. Furthermore, we extend this idea to define the notion of the direct product of two complex fuzzy subfields and also investigate the homomorphic image and inverse image of complex fuzzy subfield.
In this paper, we introduce the idea of Q-complex fuzzy sub-ring (Q-CFSR) and discuss its various algebraic aspects. We prove that every Q-CFSR generates two Q-fuzzy sub-rings (Q-FSRs). We also present the concept of level subsets of Q-CFSR and show that level subset of Q-CFSR form sub-ring. Furthermore, we extend this idea to define the notion of the direct product of two Q-CFSR Moreover, we investigate the homomorphic image and inverse image of Q-CFSR.
A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (α,β)-complex fuzzy sets and then define α,β-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of α,β-complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.
Introduction: In this paper, we present a novel hybrid model m-polar Diophantine fuzzy N-soft set and define operations on it. Methods: We generalize the concepts of fuzzy sets, soft sets, N-soft sets, fuzzy soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets and Pythagorean fuzzy N-soft sets by incorporating our proposed model. Additionally, we define three different sorts of complements for Pythagorean fuzzy Nsoft sets and examine few outcomes which do not hold in Pythagorean fuzzy N-soft sets complements unlike to crisp set. We further discuss about (α, β, γ) -cut of m-polar Diophantine fuzzy N-soft sets and their properties. Lastly, we prove our claim that the defined model is a generalization of soft set, N-soft set, fuzzy N-soft set, intuitionistic fuzzy N soft set and Pythagorean fuzzy N-soft set. Results: m-polar Diophantine fuzzy N-soft set is more efficient and an adaptable model to manage uncertainties as it also overcome drawbacks of existing models which are to be generalized. Conclusion: We introduced novel concept of m-polar Diophantine fuzzy N-soft sets (MPDFNS sets).
In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism.
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