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The enumeration of fuzzy subgroups is a complex problem. Several authors have computed results for special instances of groups. In this paper, we present a novel algorithm that is designed to enumerate the fuzzy subgroups of a finite group. This is achieved through the computation of maximal chains of subgroups. This approach is also useful for writing a program to compute the number of fuzzy subgroups. We provide further elucidation by computing the number of fuzzy subgroups of the groups Q8 and D8. [ABSTRACT FROM AUTHOR]

The dual hesitant fuzzy linguistic term set (DHFLTS) is defined by two functions that express the grade of membership and the grade of non-membership using a set of linguistic terms. In the present work, we first quote an example to point out that the existing complement operation of DHFLTS is on the wrong track. Meanwhile, we redefine this operation to fill the holes in the existing ones. Next, the notion of information energy under a dual hesitant fuzzy linguistic background is provided in order to build the criteria weight determination model. To further facilitate the theory of DHFLTS, we propose two vector similarity measures, i.e., Jaccard and Dice similarity measures, and their weighted forms for DHFLTS. In addition, we pioneer some generalized similarity measures of DHFLTSs and indicate that the Dice similarity measures are particular instances of the generalized similarity measures for some parameter values. Afterward, the similarity measures-based model with unknown weight information under the background of dual hesitant fuzzy linguistic environment is constructed. Lastly, an illustrated example is included to validate the method's application, along with sensitivity analysis and comparative analysis, demonstrating the practicality and validity of its results. [ABSTRACT FROM AUTHOR]

Hussain, Yau, and Zuo introduced the Lie algebra L k (V) from the derivation of the local algebra M k (V) : = O n / (g + J 1 (g) + ⋯ + J k (g)) . To find the dimension of a newly defined algebra is an important task in order to study its properties. In this regard, we compute the dimension of Lie algebra L 5 (V) and justify the sharp upper estimate conjecture for fewnomial isolated singularities. We also verify the inequality conjecture: δ 5 (V) < δ 4 (V) for a general class of singularities. Our findings are novel and an addition to the study of Lie algebra. [ABSTRACT FROM AUTHOR]

This research article expands the formal representation of human thinking to a most generalized hybrid theory, namely, complex q -rung orthopair fuzzy N -soft set. It is able to capture a great deal of graded imprecision and vagueness, which so often appear together in human interpretations. This model renders a parameterized mathematical tool for the ranking-based fuzzy modeling of two-dimensional paradoxical data. To that purpose, the proposed theory integrates complex q -rung orthopair fuzzy sets with the parametric structure of N -soft sets. The framework that arises captures information beyond the confined space of complex intuitionistic fuzzy N -soft sets and complex Pythagorean fuzzy N -soft sets, with the assistance of a parameter q. We establish the basic set-theoretical operations of this model and prove some of its fundamental properties. The Einstein and other elementary algebraic operations on complex q -rung orthopair fuzzy N -soft values shall be introduced to broaden the mathematical toolbox of this field. Its relationships with contemporary approaches shall demonstrate its outstanding flexibility. Moreover, we establish two competent multicriteria decision-making algorithms that capture the nuances of periodical inconsistent data. Their feasibility shall be demonstrated with an explicit application to the selection of optimum aerospace technology required for the economic development of the Mexican space agency. A comparative analysis of both strategies with the prevailing techniques substantiates their rationality. In addition, we illustrate this comparative study with an explicative bar chart that shows the compatibility of their outcomes. Finally, we examine the functionality of the proposed model and compare it with alternative theories. [ABSTRACT FROM AUTHOR]

In this paper, we set forth a framework for solving a multiattribute group decision-making (MAGDM) problem, namely, the selection of a firm for participation in a Saudi oil refinery project in Pakistan. This project will prove a key success factor for the economic growth of Pakistan due to its enormous economic impact on the energy sector, industrial development, commerce, transportation, and so on. This multiplicity justifies that several intricate components comprising both intrinsic and external attributes should be adequately evaluated for the selection of such a firm, that is, the formulation of this question as a MAGDM problem. Nonbinary evaluation with two-dimensional ambiguity and uncertainty in the parameters are general concerns in modern literature, and they fit into this problem. Within this context, one of the most superior and amenable theories (complex spherical fuzzy N -soft sets, henceforth C S F N S f S s) shall be used to formulate a new comprehensive method, known as complex spherical fuzzy N -soft-VIKOR (C S F N S f -VIKOR) method. According to the general spirit of the benchmark technique, the normalized Euclidean distances and the weights of the attributes are jointly handled, and as consequence, two main features ("maximum group utility" and "minimum individual regret") are acquired. The coefficient strategy with reference to group utility measure and individual regret measure of opponents are employed for the compromise measure. Armed with this novel tool, we single out the most feasible firm according to the preference order of the alternatives examined by the decision-makers on the subject of linear normalized weights of experts and attributes. Furthermore, a comparative analysis justifies the C S F -VIKOR method, and some results prove its capabilities and validity. Moreover, a sensitivity test certifies the stability of the proposed method. [ABSTRACT FROM AUTHOR]