In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ...ring with nonzero identity, denoted by Γ(R), is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if xy=0. Here, we classify the local rings with genus three zero-divisor graphs.
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The concept of complex Pythagorean fuzzy set (CPFS) is recent development in the field of fuzzy set (FS) theory. The significance of this concept lies in the fact that this theory assigned membership ...grades ψ and non-membership grades ˆψ from unit circle in plane, i.e., in the form of a complex number with the condition (ψ) 2 + (ˆψ) 2 ≤ 1 instead from 0, 1 interval. This is an expressive technique for dealing with uncertain circumstances. The aim of this study is to proceed the classification of the unique framework of CPFS in algebraic structure that is field theory and examine its numerous algebraic features. Also, we initiate the important examples and results of certain field. Furthermore, we illustrate that every complex Pythagorean fuzzy subfield (CPFSF) generates two Pythagorean fuzzy subfields (PFSFs).We also prove many useful algebraic aspects of this notion for a CPFSF. Moreover, we demonstrate that intersection of two complex Pythagorean fuzzy subfields (CPFSFs) is also CPFSF. Additionally, we discuss the novel idea of level subsets of CPFSFs and demonstrate that level subset of CPFSF form subfield. Additionally, we improve the application of this theory to show the concept of the direct product of two CPFSFs is also a CPFSF and produce several novel results on direct product of CPFSFs. Finally, we explore the homomorphic images and inverse images of CPFSFs.
This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by A which contains a nontrivial projection with a unit I. ...Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras.
A Fermatean fuzzy set (FRFS) is the extension of a fuzzy set, an intuitionistic fuzzy set, and a Pythagorean fuzzy set, and is used in different fields. Unlike other fuzzy structures, the sum of ...cubes of membership grades in FRFSs approximates a unit interval, increasing uncertainty. In this study, we intend to provide unique operational rules and aggregation operators (AOs) inside a Fermatean fuzzy environment. To develop a fair remedy for the membership degree and non-membership degree features of “Fermatean fuzzy numbers (FRFNs)”, our solution introduces new neutral or fair operating principles, which include the concept of proportional distribution. Based on the suggested operating principles, we provide the “Fermatean fuzzy fairly weighted average operator and the Fermatean fuzzy fairly ordered weighted averaging operator”. Our suggested AOs provide more generalized, reliable, and exact data than previous techniques. Combining the recommended AOs with multiple decision-makers and partial weight information under FRFSs, we also devised a technique for “multi-criteria decision-making”. To illustrate the application of our novel method, we provide an example of the algorithm’s effectiveness in addressing decision-making challenges.
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Data mining evaluation is very critical in the sense that it determines how well a classification model performs and how well it can generate accurate predictions on brand-new, unexplored data. It is ...especially important for classification tasks. There are several methods for evaluating classification models, and the choice of evaluation strategies depends on the particular situation, the available data, and the desired outcomes. The notion of a T-bipolar soft set (TBSS) is a valuable parameterization tool and is closer to the concept of bipolarity. Moreover, algebraic structures like groups, rings, and modules, etc., are basic tools that can be helpful not only in mathematics but also in other scientific areas due to their symmetric properties. In this article, based on the novelty of TBSS and the characteristics of rings, we have generalized these two notions to deliver and introduce the notion of T-bipolar soft rings (TBSRs). Additionally, the concepts of AND product, OR product, extended union, extended intersection, restricted union, and restricted intersection for two TBSRs is introduced, and the related results are conferred. To support these proposed notions, we have delivered examples related to these ideas. For the applicability of the developed approach, an algorithm is defined based on the delivered approach. An illustrative example regarding the classification of data mining techniques is developed to show the applications of the introduced work. We can see that there are four alternatives, and their score values are, respectively, given by −4, 42, 0, and −32. Based on these results, we can evaluate the best data mining technique. So, the defined algorithm makes it easy for us to classify the data mining techniques. Further asymmetric data are frequently employed for selecting the best alternative in decision-making problems because the information regarding alternatives is not necessarily always symmetric. Therefore, asymmetric information can be discussed using these proposed concepts.
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The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations ...and describe the structure of the quotient ring S/P, where S is any ring and P is a prime ideal of S. The symmetricity of additive mappings allows us to transfer ring theory results to functional analyses, particularly to C∗-algebras. Precisely, we describe the structures of C∗-algebras via symmetric additive mappings.
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The theory of complex Pythagorean fuzzy set (CPFS) has been already interpreted. This theory states that in both polar and Cartesian form the degree of membership and degree of non-membership are ...located in a complex plane's unit disc. However the Cartesian presentation has some drawbacks for instance, this theory can't consider the full belongingness <inline-formula> <tex-math notation="LaTeX">\left ({{ 1+\iota 1, 0+\iota 0 }}\right) </tex-math></inline-formula> of the element because this value is not located in the unit disc of a complex plane but in a unit square of a complex plane. Hence, with the existing Cartesian form of CPFS, the full belongingness of the element can't be described. Further, in polar form, the degrees of membership and non-membership are interpreted by amplitude and phase terms. Where the amplitude acts similar as they act in the Pythagorean fuzzy set (PFS) while phase terms show the periodicity, direction, or position of the element in a set. However, this description of degrees of membership and non-membership is confined to the polar structure and is inappropriate for inclusion in logical operations accompanied by CPFS in Cartesian coordinates. Therefore, in this article, we devise a theory of CPFS in Cartesian coordinates, where both degrees of membership and non-membership are in Cartesian form located in a complex plane's unit disc and containing real and imaginary terms. These terms are fuzzy functions and carry fuzzy information. We also establish a few critical and basic operations for CPFS in Cartesian coordinate and then discuss some aggregation operators (AOs) and a multi-attribute decision-making (MADM) method within CPFS. After that, we employ the established theory in the field of visualization technology to reveal the applicability and requirement of the proposed theory. We interpreted the comparison of the deduced theory with certain prevailing theories to portray the supremacy of this work.
The practice of adding new features or changing current features to enhance a machine-learning model's performance is known as feature engineering. It increases the prediction potential of machine ...learning and aids in revealing the data's underlying patterns. Different soft structures can be utilized to reach a decision where we have to decide the best alternative among the given choices. The structure of a ring is an algebraic structure that plays a vital role due to its characteristics. Moreover, a soft set is a valuable structure that can consider the parameterization tool. Also T-bipolar soft set is a parameterization tool that can consider the positive and negative aspects. Based on these observations we have developed the theory of lattice-ordered T-bipolar soft rings (LOTBSRs) and anti-lattice-ordered T-bipolar soft rings (ALOTBSRs). Moreover, we have defined the notions of OR product, extended union, and restricted union for LOTBSRs. Furthermore, the ideas of AND product, restricted intersection, and extended intersection are defined. To analyze the whole theory, we have proved some results related to these ideas. To construct the applications part of these developed notions, we have defined an algorithm and utilized these ideas in the decision-making scenarios for the classification of feature engineering techniques. In the end, we have some conclusion remarks.
The selection of the proper P2P energy trading platform is a complicated multi-attribute decision-making (MADM) dilemma that involves evaluating different alternatives against various attribute. ...Traditional MADM techniques often fail to capture the bipolarity of certain attribute, where positive and negative aspects are simultaneously present. This duality of attribute therefore requires a more advance method of modeling and decision making (DM). The bipolar fuzzy set (BFS) framework presents a possible research gap-filling solution by enabling the consideration of both positive and negative information associated with each attribute at the same time. This article aims to use BFS to create a MADM technique that will be able to model the bipolarity of the criteria, and provide a structured approach for the selection of the most suitable P2P energy trading platform. Further, this article contains various aggregation operators within BFS based on Aczel-Alsina (AA) t-norm and t-conorm, which play a critical role in the proposed MADM approach. After that, the case study, "Selection of Energy Trading Platform for P2P Energy Trading" is investigated by employing the invented approach. In the end, the advantages and dominance of the inaugurated work over some of the prevailing literature are demonstrated through comparative analysis.
Let n≥1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Zn, ...where n=pα,p2q,p2q2,pqr,p3q,p2qr, and pqrs for the different prime integers p,q,r, and s. Moreover, we construct M-polynomials and CoM-polynomials using the PIS-graph structure of Zn to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and CoM-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory.
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