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One of the most helpful fuzzy set extensions for dealing with information uncertainties is an <inline-formula> <tex-math notation="LaTeX">$\text{n}^{th}$ </tex-math></inline-formula> power root fuzzy set. In light of this, in this paper, we define an n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy set, which is a new type of fuzzy set extension and introduce their relationship with n-rung orthopair fuzzy set, <inline-formula> <tex-math notation="LaTeX">$\text{n}^{th}$ </tex-math></inline-formula> power root fuzzy set and n,m-rung orthopair fuzzy set. There is a symmetry between the values of this membership and non-membership functions. To broaden the scope of the decision-making problems, any power function scales are used here. When disputing the symmetry between two or more objects, the innovative concept of an n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, as well as Pythagorean fuzzy set. Then, for the n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy sets, we present the essential set of operations as well as their varied characteristics. In addition, we develop the idea of n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy topology and study its fundamental properties. Furthermore, we describe the concept of disconnected n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy sets after introducing separated n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy sets. Moreover, we thoroughly investigate and characterize n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy continuous maps. Also we build <inline-formula> <tex-math notation="LaTeX">$T_{0}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$T_{1}$ </tex-math></inline-formula> in n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy topology and find their connections. Finally, we construct a new concept of relation in n,<inline-formula> <tex-math notation="LaTeX">$\text{m}^{th}$ </tex-math></inline-formula> power root fuzzy set, and based on sufficient experience, we provide the candidate&#x2019;s decision-making technique via the proposed relation to determine the suitability of companies to applicants.

n,mth Power Root Fuzzy Set and Its Applications to Topology and Decision-Making

n,m^th Power Root Fuzzy Set and Its Applications to Topology and Decision-Making