Idempotent Property Research Articles

On Some Classes of Regular Semigroups

In this paper we study and analyze the relationships between some special classes of regular semigroups, namely: Generalized inverse semigroups, Orthodox semigroups and locally inverse semigroups based on their idempotent properties. We then use an example to prove that every locally inverse semigroup is not a generalized inverse semigroup, thus validating the statement; “A generalized inverse semigroup has the property of being locally inverse. However, not every locally inverse semigroup is a generalized inverse semigroup.”

Complex Pythagorean neutrosophic normal interval-valued set with an aggregation operators using score values

The complex Pythagorean neutrosophic normal interval-valued set approach solves the multiple-attribute decision-making problem. We introduce the new concepts such as complex Pythagorean neutrosophic normal interval-valued weighted averaging, complex Pythagorean neutrosophic normal interval-valued weighted geometric, complex generalized Pythagorean neutrosophic normal interval-valued weighted averaging and complex generalized Pythagorean neutrosophic normal interval-valued weighted geometric operator. We demonstrate that complex Pythagorean neutrosophic normal interval-valued set satisfy algebraic structures such as associative, idempotent, bounded, commutative and monotonic properties. Additionally, we develop algorithm and flowchart that solve problems using these operators. Examples of using enhanced score values and accuracy values in real-world environments are provided in this paper. Artificial intelligence refers to the simulation or approximation of human intelligence in machines. Its goals include computer enhanced learning, reasoning and perception. Artificial intelligence is being used today across different industries, from finance to healthcare. Agricultural robots have been described as being highly dependent on computer and machine tool technology. Four factors can be used to evaluate the quality of a robotics system: the controller’s sophistication, the software efficiency, the maximum moment of inertia, and the manufacturer’s reliability. The best alternative can be determined by comparing expert opinions to the criteria. Therefore, the parameter ∇ has a very significant impact on the results of the model. This comparison aims to prove that the models under consideration are valid and valuable by comparing them with the available and proposed models. In conclusion, the value of ∇ significantly impacts the model performance. Based on the comparison and sensitivity analysis, we conclude that the proposed aggregation operation is superior and more reliable than the existing one. The criteria were compared to the most appropriate options based on expert assessments.

Complex Pythagorean Normal Interval-Valued Fuzzy Aggregation Operators for Solving Medical Diagnosis Problem

This paper presents a new methodology for solving multiple-attribute decision-making problems (MADMs) using a complex Pythagorean normal interval-valued fuzzy set (CPNIVFS), which is an extended concept of a complex Pythagorean fuzzy set. Four types of different aggregating operations (AOs), including CPNIVF weighted averaging (CPNIVFWA), CPNIVF weighted geometric (CPNIVFWG), generalized CPNIVFWA (CGPNIVFWA), and generalized CPNIVFWG (CGPNIVFWG), are discussed. The scoring function, accuracy function, and operational laws of the CPNIVFS are defined. Algebraic structures, such as associative, distributive, idempotent, bounded, commutativity, and monotonicity properties, are also shown to be satisfied by complex Pythagorean normal interval-valued fuzzy numbers. Furthermore, an algorithm is proposed to solve the MADM problems based on the defined AOs. The proposed approach is then used for a medical diagnosis problem about brain tumors because computer science and machine tool technology are among the most important components of brain tumor research. The five types of brain tumors diagnosed in these patients are gliomas, meningiomas, metastases, embryonal tumors, and ependymomas. Several types of treatments are available, which are often combined as part of an overall treatment plan. Brain tumors can be treated in various ways, including surgery, radiation therapy, chemotherapy, immunotherapy, and clinical trials. Based on the comparisons and options gathered, the most suitable treatment can be chosen. In this regard, it is evident that the value of the integer ⅁\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Game $$\\end{document} plays a significant role in determining the model. The candidate models under consideration can be validated by comparing them with the previously proposed ones. The proposed technique is compared with the existing method to demonstrate its superiority and validity, and the results conclude that the former is more reliable and effective than the latter. Finally, the criteria are evaluated by expert assessments to determine the most appropriate options.

On Circular q-Rung Orthopair Fuzzy Sets with Dombi Aggregation Operators and Application to Symmetry Analysis in Artificial Intelligence

Circular q-rung orthopair fuzzy sets (FSs) were recently considered as an extension of q-rung orthopair FSs (q-ROFSs), circular intuitionistic FSs (Cir-IFSs), and circular Pythagorean FSs (Cir-PFSs). However, they are only considered for some simple algebraic properties. In this paper, we advance the work on circular q-ROFSs (Cirq-ROFSs) in Dombi aggregation operators (AOs) with more mathematical properties of algebraic laws. These include the circular q-rung orthopair fuzzy (Cirq-ROF) Dombi weighted averaging (Cirq-ROFDWA), Cirq-ROF Dombi ordered weighted averaging (Cirq-ROFDOWA), Cirq-ROF Dombi weighted geometric (Cirq-ROFDWG), and Cirq-ROF Dombi ordered weighted geometric (Cirq-ROFDOWG) operators. Additionally, we present the properties of idempotency, monotonicity, and boundedness for the proposed operators. In the context of artificial intelligence, symmetry analysis plays a significant and efficient role that can refer to several aspects. Thus, to compute the major aspect, we identify the multi-attribute decision-making (MADM) technique based on the proposed operators for Cirq-ROF numbers (Cirq-ROFNs) to enhance the worth of the evaluated operators. Finally, we use some existing techniques for comparison to our results to show the validity and supremacy of the proposed method.

Selection process based on new building construction work using square root vague sets and their aggregated operators

This study discusses novel approaches for solving multi-attribute decision-making problems using logarithmic square root vague sets. Logarithmic square root vague sets can be considered as an extension of vague sets and square root fuzzy sets provides a more profitable method to express the uncertainties in the data to deal with decision making problems. We developed some novel logarithmic square root vague sets, which can overcome the weaknesses of algebraic operations and capture the relationship between various square root vague sets and vague sets. The purpose of this study is to examine several aggregation operators using logarithmic square root vague sets. The weighted logarithmic averaging distance and weighted logarithmic geometric distance operators are distance measure that is based on aggregating model. We discuss the concepts of logarithmic square root vague weighted averaging, logarithmic square root vague weighted geometric, generalized logarithmic square root vague weighted averaging and generalized logarithmic square root vague weighted geometric operators. We communicate that algebraic structures such as associative, distributive, idempotent, bounded, commutativity and monotonicity properties are satisfied by logarithmic square root vague numbers. We discusses some mathematical properties of logarithmic square root vague sets, as well as the Hamming distance and Euclidean distance measures. It measures the similarity of two strings of information by measuring the Hamming distance. An analysis of the proposed operators is presented, along with alternative formulations and families, as well as their main properties. The weighting vector is characterized by multiple classical measures and we propose alternative solutions for dealing with the logarithmic properties of the operators. As a result of studying the weighting vectors of the operators, we present generalizations of them. We examine vague sets with important properties using algebraic operations in more detail. Furthermore, we develop an algorithm to solve multi-attribute decision-making problems using these operators. We provide real-life examples to illustrate how enhanced Euclidean distances, Hamming distances and score values can be applied in real-world situations. To show the superiority and the validity of the proposed aggregation operations, we compared it with the existing method, and concluded from the comparison and sensitivity analysis that our proposed technique is more effective and reliable. Based on the expert assessments, we compared the criteria with the most appropriate options.

Applications Aczel-Alsina t-norm and t-conorm for the assessment of fire extinguishers using Pythagorean fuzzy information

Prioritization is usually required in problems involving multi-attribute group decision-making (MAGDM). Several strategies and procedures have been introduced in fuzzy systems to apply prioritization. This study examines the MAGDM problem in a Pythagorean fuzzy (PF) setting with varying amounts of demand for specialists and attributes. We regard the novel Aczel Alsina aggregation operators (AOs) as the most addition to fuzzy mathematics that can deal with uncertainties significantly. We suggest a few PF AOs based on Aczel Alsina t-norm and t-conorm, including the PF-prioritized Aczel Alsina averaging (PFPAAA) and PF-prioritized Aczel Alsina geometric (PFPAAG) operators. It is proven that these AOs fulfil the aggregation criteria by investigating the properties of monotonicity, boundedness, and idempotency. The weights for prioritization are derived from the knowledge of experts, and the proposed operators can capture the phenomenon of prioritization among the aggregated arguments. The proposed AOs are then applied to assess fire extinguishers using a MAGDM technique. The importance of PFPAAA and PFPAAG operators is verified by comparing the proposed AOs with other well-known AOs.

An Innovative Decision Model Utilizing Intuitionistic Hesitant Fuzzy Aczel-Alsina Aggregation Operators and Its Application

The intuitionistic hesitant fuzzy set is a significant extension of the intuitionistic fuzzy set, specifically designed to address uncertain information in decision-making challenges. Aggregation operators play a fundamental role in combining intuitionistic hesitant fuzzy numbers into a unified component. This study aims to introduce two novel approaches. Firstly, we propose a three-way model for investors in the business domain, which utilizes interval-valued equivalence classes under the framework of intuitionistic hesitant fuzzy information. Secondly, we present a multiple-attribute decision-making (MADM) method using various aggregation operators for intuitionistic hesitant fuzzy sets (IHFSs). These operators include the IHF Aczel–Alsina average (IHFAAA) operator, the IHF Aczel–Alsina weighted average (IHFAAWAϣ) operator, and the IHF Aczel–Alsina ordered weighted average (IHFAAOWAϣ) operator and the IHF Aczel–Alsina hybrid average (IHFAAHAϣ) operators. We demonstrate the properties of idempotency, boundedness, and monotonicity for these newly established aggregation operators. Additionally, we provide a detailed technique for three-way decision-making using intuitionistic hesitant fuzzy Aczel–Alsina aggregation operators. Furthermore, we present a numerical case analysis to illustrate the pertinency and authority of the esteblished model for investment in business. In conclusion, we highlight that the developed approach is highly suitable for investment selection policies, and we anticipate its extension to other fuzzy information domains.

On Hybrid Pure Hyperideals in Ordered Hypersemigroups

In this paper, the concepts of hybrid pure hyperideals in ordered hypersemigroups are introduced and some algebraic properties of hybrid pure hyperideals are studied. We characterize weakly regular ordered hypersemigroups in terms of hybrid pure hyperideals. Finally, we introduce the concepts of hybrid weakly pure hyperideals and prove that the hybrid hyperideals are hybrid weakly pure hyperideals if such hybrid hyperideals satisfy the idempotent property.

Frank Prioritized Aggregation Operators and WASPAS Method Based on Complex Intuitionistic Fuzzy Sets and Their Application in Multi-Attribute Decision-Making

Complex intuitionistic fuzzy (CIF) information covers the degree of membership and the degree of non-membership in the form of polar coordinates with a valuable and dominant characteristic where the sum of the real parts (the same rule for the imaginary parts) of the pair must be contained in the unit interval. In this paper, we first derive the Frank operational laws for CIF information and then examine the prioritized aggregation operators based on Frank operational laws for managing the theory of CIF information. These are the CIF Frank prioritized averaging (CIFFPA) operator, the CIF Frank prioritized ordered averaging (CIFFPOA) operator, the CIF Frank prioritized geometric (CIFFPG) operator, and the CIF Frank prioritized ordered geometric (CIFFPOG) operator with properties of idempotency, monotonicity, and boundedness. Furthermore, we derive the WASPAS (weighted aggregates sum product assessment) under the consideration or presence of the CIF information and try to justify it with the help of a suitable example. Additionally, we illustrate some numerical examples in the presence of the MADM (multi-attribute decision-making) procedures for evaluating the comparison between the proposed operators with some well-known existing operators to show the validity and worth of the proposed approaches.

Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties

Algorithms for computing congruence closure of ground equations over uninterpreted symbols and interpreted symbols satisfying associativity and commutativity (AC) properties are proposed. The algorithms are based on a framework for computing a congruence closure by abstracting nonflat terms by constants as proposed first in Kapur's congruence closure algorithm (RTA97). The framework is general, flexible, and has been extended also to develop congruence closure algorithms for the cases when associative-commutative function symbols can have additional properties including idempotency, nilpotency, identities, cancellativity and group properties as well as their various combinations. Algorithms are modular; their correctness and termination proofs are simple, exploiting modularity. Unlike earlier algorithms, the proposed algorithms neither rely on complex AC compatible well-founded orderings on nonvariable terms nor need to use the associative-commutative unification and extension rules in completion for generating canonical rewrite systems for congruence closures. They are particularly suited for integrating into the Satisfiability modulo Theories (SMT) solvers. A new way to view Groebner basis algorithm for polynomial ideals with integer coefficients as a combination of the congruence closures over the AC symbol * with the identity 1 and the congruence closure over an Abelian group with + is outlined.

Information Fusion Model of Group Decision Making Based on a Combinatorial Ordered Weighted Average Operator

Aiming at the problem that the existing aggregation operators only consider the weight of the data itself or the weight of the data position when aggregating data, a group decision information fusion model based on the combinatorial ordered weighted average operator is proposed, which considers the incompleteness, uncertainty and plurality of the decision information, and effectively improves the accuracy of group consensus. In this paper, an interval intuitionistic fuzzy combinatorically weighted average operator is proposed, which embodies the weight of the set data itself and the weight of data position, and can adjust the importance of the two weights in the operator through parameters, and prove that the operator has the properties of boundedness, monotonicity, idempotency and permutation invariance. Secondly, a position weight solving model of interval intuitionistic fuzzy combinatorial ordered weighted average operator is established, and the position weight is solved with the help of cross-entropy and Orness measures. Finally, a group decision information fusion process based on IVIFCOWA operators is designed, and the decision maker weight information is obtained according to the decision matrix, and the comprehensive decision matrix is obtained by fusion of IVIFCOWA operators, and then the mean similarity is obtained. Through specific case analysis, it is proved that the proposed operator can make full use of the decision data, and can better reach group consensus after data fusion than the existing operator.

Prioritization of Strategies of Digital Transformation of Supply Chain Employing Bipolar Complex Fuzzy Linguistic Aggregation Operators

Digitalization has addressed all parts of organizations together with supply chain management. Innovations like installed RFID, GPS, and sensors have assisted organizations with changing their current conventional supply chain framework into more spry, adaptable, open, and cooperative computerized models. A lot of organizations want to get the advantages of digitalization, particularly in the supply chain But here, the biggest problem, that the organizations are facing is the ambiguities, i.e. how to select and evaluate the best strategy for the supply chain digital transformation (SCDT), on which criteria or attribute they can assess the strategies of SCDT, etc. Multi-attribute decision-making (MADM) procedure is the finest technique to evaluate and find out the finest strategy of the SCDT. To handle these ambiguities and for making the MADM technique, this study first establishes the concept of bipolar complex fuzzy linguistic sets (BCFLSs) and its essential properties. Furthermore, we scrutinize average and geometric AOs for BCFLS such as bipolar complex fuzzy linguistic weighted averaging (BCFLWA) and bipolar complex fuzzy linguistic weighted geometric (BCFLWG) operators along with their idempotency, boundedness, and monotonicity properties. After that, to exhibit the usefulness of our established notion and AOs in real life and the selection and evaluation of the strategy of digital transformation of the supply chain, we interpret a DM technique and explore a numerical example. To exhibit the supremacy of the defined BCFLS and defined AOs we compare them with certain prevailing conceptions.

Novel Complex Pythagorean Fuzzy Sets under Aczel–Alsina Operators and Their Application in Multi-Attribute Decision Making

Aggregation operators (AOs) are utilized to overcome the influence of uncertain and vague information in different fuzzy environments. A multi-attribute decision-making (MADM) technique plays a vital role in several fields of different environments such as networking analysis, risk assessment, cognitive science, recommender systems, signal processing, and many more domains in ambiguous circumstances. In this article, we elaborated the notion of Aczel–Alsina t-norm (TNM) and t-conorm (TCNM) under the system of complex Pythagorean fuzzy (CPyF) sets (CPyFSs). Some basic operational laws of Aczel–Alsina TNM and TCNM are established including Aczel–Alsina sum, product, scalar multiplication, and power operations based on CPyFSs. We established several AOs of CPyFSs such as CPyF Aczel–Alsina weighted average (CPyFAAWA), and CPyF Aczel–Alsina weighted geometric (CPyFAAWG) operators. The proposed CPyFAAWA and CPyFAAWG operators are symmetric in nature and satisfy the properties of idempotency, monotonicity, boundedness and commutativity. To solve an MADM technique, we established an illustrative example to select a suitable candidate for a vacant post in a multinational company. To see the advantages of our proposed AOs, we compared the results of existing AOs with the results of newly established AOs.

A Multi-Attribute Decision-Making Approach for the Analysis of Vendor Management Using Novel Complex Picture Fuzzy Hamy Mean Operators

Vendor management systems (VMSs) are web-based software packages that can be used to manage businesses. The performance of the VMSs can be assessed using multi-attribute decision-making (MADM) techniques under uncertain situations. This article aims to analyze and assess the performance of VMSs using MADM techniques, especially when the uncertainty is of complex nature. To achieve the goals, we aim to explore Hany mean (HM) operators in the environment of complex picture fuzzy (CPF) sets (CPFSs). We introduce CPF Hamy mean (CPFHM) and CPF weighted HM (CPFWHM) operators. Moreover, the reliability of the newly proposed HM operators is examined by taking into account the properties of idempotency, monotonicity, and boundedness. A case study of VMS is briefly discussed, and a comprehensive numerical example is carried out to assess VMSs using the MADM technique based on CPFHM operators. The sensitivity analysis and comprehensive comparative analysis of the proposed work are discussed to point out the significance of the newly established results.

On Tripolar Fuzzy Pure Ideals in Ordered Semigroups

Tripolar fuzzy sets are a concept that deals with tripolar information. This idea is a generalization of bipolar and intuitionistic fuzzy sets. In this paper, the notions of tripolar fuzzy pure ideals in ordered semigroups are introduced, and some algebraic properties of tripolar fuzzy pure ideals are studied. We obtain some characterizations of weakly regular ordered semigroups in terms of tripolar fuzzy pure ideals. Finally, we introduce the concepts of tripolar weakly pure ideals and prove that the tripolar fuzzy ideals are tripolar weakly pure ideals if such tripolar fuzzy ideals satisfy the idempotent property.

Discrete overlap functions: Basic properties and constructions

As a kind of emerging binary continuous aggregation operator that has been successfully applied in many practical application problems, overlap functions on the unit closed interval have been considered by scholars on different truth values sets lately. At the same time, studying aggregation operators on finite chains, especially for commonly used binary aggregation operators, is a meaningful and hot topic in the research field of aggregation operators. In this paper, we pay attention to overlap functions on finite chains, which are called discrete overlap functions. Specifically, first, we introduce the notions of discrete overlap functions on the finite chain L with n+2 elements and its arbitrary subchains along with an extended form of them. Second, we study some basic properties of discrete overlap functions on L, especially for the idempotent property, Archimedean property and cancellation law. In particular, we obtain some new properties which are different from those of the overlap functions on other truth values sets, for instance, every discrete overlap function on L takes the greatest element on L as the neutral element. Third, we discuss the construction methods of discrete overlap functions on L. Finally, it is worth mentioning that the results obtained in this paper provide a theoretical basis and more possibilities for the potential applications of overlap functions in other fields besides their known applications, especially for the situation of that the reasoning of experts are described by linguistic terms or labels, such as in expert systems, fuzzy control and etc.

New results on Congruence Boolean Lifting Property

The Lifting Idempotent Property (LIP) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures (MV-algebras, commutative l-groups, BL-algebras, bounded distributive lattices, residuated lattices, etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property (CBLP) for congruence modular algebras). A lifting ideal of a ring R is an ideal of R fulfilling LIP. In a recent paper, Tarizadeh and Sharma obtained new results on lifting ideals in commutative rings. The aim of this paper is to extend an important part of their results to congruences with CBLP in semidegenerate congruence modular algebras. The reticulation of such algebra will play an important role in our investigations (recall that the reticulation of a congruence modular algebra A is a bounded distributive lattice L(A) whose prime spectrum is homeomorphic with Agliano's prime spectrum of A). Almost all results regarding CBLP are obtained in the setting of semidegenerate congruence modular algebras having the property that the reticulations preserve the Boolean center. The paper contains several properties of congruences with CBLP. Among the results we mention a characterization theorem of congruences with CBLP. We achieve various conditions that ensure CBLP. Our results can be applied to a lot of types of concrete structures: commutative rings, l-groups, distributive lattices, MV-algebras, BL-algebras, residuated lattices, etc.

Zero-sum triangles for involutory, idempotent, nilpotent and unipotent matrices

In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. Some of the generated integer triangles have been found by other methods, but most of them are newly discovered, and many combinatorial identities can be found with them. The results may also interest the economists for trading analysis and simulation.

A Novel Spherical Fuzzy Rough Aggregation Operators Hybrid with TOPSIS Method and Their Application in Decision Making

Industrial control system (ICS) attacks are usually targeted attacks that use the ICS entry approach to get a foothold within a system and move laterally throughout the organization. In recent decades, powerful attacks such as Stuxnet, Duqu, Flame, and Havex have served as wake-up calls for industrial units. All organizations are faced with the rise of security challenges in technological innovations. This paper aims to develop aggregation operators that can be used to address the decision-making problems based on a spherical fuzzy rough environment. Meanwhile, some interesting properties of idempotence, boundedness, and monotonicity for the proposed operators are analyzed. Moreover, we use this newly constructed framework to select ICS security suppliers and validate its acceptability. Furthermore, a different test has been performed based on a new operator to strengthen the suggested approach. Additionally, comparative analysis based on the novel extended TOPSIS method is presented to demonstrate the superiority of the proposed technique. The results show that the conventional approach has a larger area for information representation, better adaptability to the evaluation environment, and higher reliability of the evaluation results.

Interval‐Valued q‐Rung Orthopair Fuzzy QUALIFLEX Decision Analysis Method with Dombi Operators

As a new and useful decision tool, interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS), has strong ability to address uncertain or imprecision information in practical decision‐making processes. This paper develops a new interval‐valued q‐rung orthopair fuzzy QUALIFLEX decision analysis method with Dombi operators for handling the group decision‐making problems with IVq‐ROFSs data. We also conduct a case study on the health management of hypertensive patients for demonstrating the feasibility and superiority of the developed method. The main contributions of this paper are as follows: (1) we first propose interval‐valued q‐rung orthopair fuzzy Dombi operational laws and investigate their desirable properties; (2) using the developed Dombi operational laws, we develop an interval‐valued q‐rung orthopair fuzzy Dombi weighted average (IVq‐ROFDWA) operator, and we prove that the IVq‐ROFDWA operator satisfies the properties of idempotency, commutativity, boundedness, and monotonicity; (3) we introduce the concept of interval‐valued q‐rung orthopair fuzzy ideal group opinions with the help of IVq‐ROFDWA operator and propose a new weight‐determining method for identifying the weights of experts; (4) we develop a new interval‐valued q‐rung orthopair fuzzy distance measure. With the help of the developed distance measure, we calculate interval‐valued q‐rung orthopair fuzzy concordance/discordance index under each permutation, and then a new interval‐valued q‐rung orthopair fuzzy QUALIFLEX decision analysis method is developed for addressing the complex multiattribute group decision‐making problems with IVq‐ROFSs.