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The Pythagorean fuzzy environment is a modern way of depicting uncertainty. The concept of Pythagorean fuzzy semi-level subgroups of any group is described in this paper. The Pythagorean fuzzy order of an element in a Pythagorean fuzzy subgroup is introduced and established various algebraic attributes. The relation between the Pythagorean fuzzy order of an element of a group and the order of that group is established. The Pythagorean fuzzy normalizer and Pythagorean fuzzy centralizer of Pythagorean fuzzy subgroups are discussed. Further, the concept of Pythagorean fuzzy quotient group and the index of a Pythagorean fuzzy subgroup are defined. Finally, a framework is developed for proving Lagrange's theorem in Pythagorean fuzzy subgroups. [ABSTRACT FROM AUTHOR]

Fuzzy set is a modern tool for depicting uncertainty. This paper introduces the concept of fuzzy sub e-group as an extension of fuzzy subgroup. The concepts of identity and inverse are generalized in fuzzy sub e-groups. Every fuzzy subgroup is proven to be a fuzzy sub e-group, but the converse is not true. Various properties of fuzzy sub e-groups are established. Moreover, the concepts of proper fuzzy sub e-group and super fuzzy sub e-group are discussed. Further, the concepts of fuzzy e-coset and normal fuzzy sub e-group are presented. Finally, we describe the effect of e-group homomorphism on normal fuzzy sub e-groups. [ABSTRACT FROM AUTHOR]

A Pythagorean fuzzy set (PFS) is a very efficient and powerful tool for handling uncertainty and vagueness. In this paper, we present the notion of (α,β) -Pythagorean fuzzy set (PFS) as a generalisation of Pythagorean fuzzy set (PFS). We propose a new fuzzy subgroup, called (α,β) -Pythagorean fuzzy subgroup (PFSG). We investigate properties of our proposed fuzzy subgroup. We present the concept of (α,β) -Pythagorean fuzzy coset (PFC) and (α,β) -Pythagorean fuzzy normal subgroup (PFNSG). Furthermore, we define (α,β) -Pythagorean fuzzy level subgroup (PFLSG) and establish properties of it. [ABSTRACT FROM AUTHOR]

In this note, we show by examples that Definitions 3.3, 3.5, Propositions 3.9, 3.10 and Theorem 3.17 in the paper by Akram and Dudek (Neural Comput Appl 21(1):197-205, 2012) contain some flaws, and then, we presented the updated results. Hence, we introduce generalized regular bipolar fuzzy graphs. [ABSTRACT FROM AUTHOR]

This paper deals with fuzzy permutation graph which is introduced as a fuzzified model of a crisp permutation graph. We have defined two types of complements of a fuzzy permutation graph viz. p-complement and f-complement. We have considered various examples to encounter the existence and non-existence of the f-complement graph as a fuzzy permutation graph and established a sufficient condition for the existence of the same. Also, we have calculated the possible number of fuzzy permutation graphs that can represent the f-complement of a fuzzy permutation graph for a given permutation depending on the edge-membership values of the fuzzy permutation graph. We have also introduced and characterized fuzzy comparability graph. Lastly, we have discussed a real-life application of fuzzy permutation graph. [ABSTRACT FROM AUTHOR]

Recently, Ghorai and Pal [12] introduced the notion of m-polar fuzzy planar graphs using the concept of m-polar fuzzy multisets. In this paper, m-polar fuzzy faces and strong m-polar fuzzy faces are defined. Then m-polar fuzzy dual graph of an m-polar fuzzy planar graph is introduced with example. Some isomorphism properties of m-polar fuzzy planar graphs are also studied. [ABSTRACT FROM AUTHOR]

Picture fuzzy sets (PFSs) can be used to handle real-life problems with uncertainty and vagueness more effectively than intuitionistic fuzzy sets (IFSs). In the process of information aggregation, many aggregation operators under PFSs are used by different authors in different fields. In this article, a multi-attribute decision-making (MADM) problem is introduced utilizing harmonic mean aggregation operators with trapezoidal fuzzy number (TrFN) under picture fuzzy information. Three harmonic mean operators are developed namely trapezoidal picture fuzzy weighted harmonic mean (TrPFWHM) operator, trapezoidal picture fuzzy order weighted harmonic mean (TrPFOWHM) operator and trapezoidal picture fuzzy hybrid harmonic mean (TrPFHHM) operator. The related properties about these operators are also studied. At last, an MADM problem is considered to interrelate among these operators. Furthermore, a numerical instance is considered to explain the productivity of the proposed operators. [ABSTRACT FROM AUTHOR]

As an extension of intuitionistic fuzzy sets, the theory of picture fuzzy sets not only deals with the degrees of rejection and acceptance but also considers the degree of refusal during a decision-making process; therefore, by incorporating this competency of picture fuzzy sets, the goal of this study is to propose a novel hybrid model called picture fuzzy soft expert sets by combining picture fuzzy sets with soft expert sets for dealing with uncertainties in different real-world group decision-making problems. The proposed hybrid model is a more generalized form of intuitionistic fuzzy soft expert sets. Some novel desirable properties of the proposed model, namely, subset, equality, complement, union and intersection, are investigated together with their corresponding examples. Two well-known operations AND and OR are also studied for the developed model. Further, a decision-making method supporting by an algorithmic format under the proposed approach is presented. Moreover, an illustrative application is provided for its better demonstration, which is subjected to the selection of a suitable company of virtual reality devices. Finally, a comparison of the initiated method is explored with some existing models, including intuitionistic fuzzy soft expert sets. [ABSTRACT FROM AUTHOR]