Idempotent Property Research Articles

Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures

The aim of this paper is to study, from an algebraic point of view, the properties of interdependencies between sets of elements (i.e., pieces of secrets, atmospheric variables, etc.) that appear in various natural models, by using the algebraic hyperstructure theory. Starting from specific examples, we first define the relation of dependence and study its properties, and then, we construct various hyperoperations based on this relation. We prove that two of the associated hypergroupoids are H v -groups, while the other two are, in some particular cases, only partial hypergroupoids. Besides, the extensivity and idempotence property are studied and related to the cyclicity. The second goal of our paper is to provide a new interpretation of the dependence relation by using elements of the theory of algebraic hyperstructures.

Interval Neutrosophic Reducible Weighted Maclaurin Symmetric Means With Internet of Medical Things (IoMt) Industry Evaluation

The Internet of Medical Things (IoMT) is a global infrastructure composing of plentiful applications and medical devices that are interconnected by ICT. In considering the problem of the IoMT industry evaluation, the requisite issue that concerns strong interaction and incertitude. The Maclaurin symmetric mean (MSM), as a resultful information concordant instrument, can capture the interrelation among multiple arguments more efficiently. The abundance of the weighted MSMs has been presented to manage the different uncertain information aggregation issues by reason that the attribute variables are frequently diverse. However, these existing weighted form of MSM operators fail to possess the fundamental properties of idempotency and reducibility. To solve the above issues, we explore the interval neutrosophic reducible weighted MSM (INRWMSM) operator and the interval neutrosophic reducible weighted dual MSM (INRWDMSM) operator. Moreover, momentous properties and some special cases of the INRWMSM and INRWDMSM operators are discussed in detail. Whereafter, we propose some multiple attribute decision making (MADM) algorithms based on INRWMSM and INRWDMSM. The availability of proposed algorithms is stated by an IoMT evaluation issue. Finally, a comparison of the developed with the existing interval neutrosophic decision making algorithms has been formed for showing their validity.

Intuitionistic fuzzy reducible weighted Maclaurin symmetric means and their application in multiple-attribute decision making

As an important information aggregation tool, the Maclaurin symmetric mean (MSM) can capture the correlation between multiple input values and has recently become a hot topic in the field of academic research. Due to the importance of the fact that attribute variables are often different, many weighted MSMs have been designed to deal with various fuzzy information aggregation problems. However, these weighted form operators do not have the properties of idempotency, i.e., the weighted average value of equivalent fuzzy numbers varies with their weights. In addition, when their weights are equal, the weighted MSMs cannot reduce to the MSM, which means they do not have reducibility. To solve these problems, in this paper, we introduce the reducible weighted MSM (RWMSM) and the reducible weighted dual MSM (RWDMSM), and we extend them to aggregate intuitionistic fuzzy information. In order to better analyze and understand the operation mechanism of the proposed weighted MSMs, we discuss several advantageous properties and special related cases of the proposed weighted MSMs. The other objective of this paper is to investigate the practice application of the proposed weighted MSMs in decision making under conditions of an intuitionistic fuzzy environment. A case study shows that the decision-making method based on the intuitionistic fuzzy RWMSM and RWDMSM can flexibly capture the correlation and reflect the decision maker’s risk preference.

Universal gradings of orders

For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order} has a universal grading, and this grading is by a {\em finite} group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.

Commentary on “New aggregation operators of single-valued neutrosophic hesitant fuzzy set and their application in multi-attribute decision making”

Liu and Luo (Pattern Anal Appl https://doi.org/10.1007/s10044-017-0635-6 ) proposed a single-valued neutrosophic hesitant fuzzy ordered weighted aggregation (SVNHFOWA) operator and used it for solving single-valued neutrosophic hesitant fuzzy multi-attribute decision making (MADM) problems. It is pertinent to mention that to show the validity of SVNHFOWA operator, Liu and Luo claimed that this operator satisfies the idempotency, commutativity and monotonicity properties. However, after a deep study, it is observed that the monotonicity property is not satisfying for the SVNHFOWA operator whereas, for the validity of an aggregation operator the monotonicity property should necessarily be satisfied. Therefore, the SVNHFOWA operator, proposed by Liu and Luo, is not valid. Since, in future, the other researchers may use this SVNHFOWA operator in their research work as well as for solving real-life single-valued neutrosophic hesitant fuzzy MADM problems. Therefore, the aim of this paper is to make the researchers aware about the fact that it is scientifically incorrect to use the SVNHFOWA operator.

Induced simplified neutrosophic correlated aggregation operators for multi-criteria group decision-making

ABSTRACTInduced Choquet integral is a powerful tool to deal with imprecise or uncertain nature. This study proposes a combination process of the induced Choquet integral and neutrosophic information. We first give the operational properties of simplified neutrosophic numbers (SNNs). Then, we develop some new information aggregation operators, including an induced simplified neutrosophic correlated averaging (I-SNCA) operator and an induced simplified neutrosophic correlated geometric (I-SNCG) operator. These operators not only consider the importance of elements or their ordered positions, but also take into account the interactions phenomena among decision criteria or their ordered positions under multiple decision-makers. Moreover, we present a detailed analysis of I-SNCA and I-SNCG operators, including the properties of idempotency, commutativity and monotonicity, and study the relationships among the proposed operators and existing simplified neutrosophic aggregation operators. In order to handle the multi-criteria group decision-making (MCGDM) situations where the weights of criteria and decision-makers usually correlative and the criterion values are considered as SNNs, an approach is established based on I-SNCA operator. Finally, a numerical example is presented to demonstrate the proposed approach and to verify its effectiveness and practicality.

Prioritized Aggregation Operators and Correlated Aggregation Operators for Hesitant 2-Tuple Linguistic Variables

The aggregation operator is a potential tool to fuse the information derived from multisources, which has been applied in group decision, combination classification and scheduling clusters successfully. To better characterize complex decision situations and capture complex opinions of decision-makers (DMs), aggregation operators are required to be explored from different viewpoints. In view of information fusion of hesitant 2-tuple linguistic variables, this paper establishes four new aggregation operators, which are called the hesitant 2-tuple linguistic prioritized weighted averaging (H2TLPWA) aggregation operator, hesitant 2-tuple linguistic prioritized weighted geometric (H2TLPWG) aggregation operator, hesitant 2-tuple linguistic correlated averaging (H2TLCA) aggregation operator, and hesitant 2-tuple linguistic correlated geometric (H2TLCG) aggregation operator, respectively. The H2TLPWA aggregation operator and H2TLPWG aggregation operator can characterize the prioritization relationship of the aggregated arguments. The H2TLCA aggregation operator and H2TLCG aggregation operator can describe dependencies between criteria in decision-making problem solving. Moreover all aggregation operation operators have the properties of idempotency, boundedness and monotonicity, and the H2TLCA aggregation operator and H2TLCG aggregation operator are also verified to be symmetric functions. In addition, the H2TLPWA aggregation operator and H2TLCA aggregation operator are employed to settle multicriteria decision-making problems with hesitant 2-tuple linguistic terms. By virtue of predefining discrete initial linguistic labels with symmetrical distribution, the detailed steps of the decision-making process with an example are given to illustrate their practicality and effectiveness.

New Operators for Aggregating Intuitionistic Fuzzy Information With Their Application in Decision Making

Aggregation of intuitionistic fuzzy information is a hot topic in Atanassov’s intuitionistic fuzzy set theory, which has attracted much interest from researchers in recent years. In this paper, a series of new aggregation operators and weighted averaging operators are proposed for aggregating intuitionistic fuzzy information. First, some basic laws for operations on intuitionistic fuzzy values are presented together with their properties. Then, we propose intuitionistic fuzzy weighted arithmetic averaging operator and intuitionistic fuzzy weighted geometric averaging operator to aggregate intuitionistic fuzzy information. Inspired by the idea of ordered weighted averaging and hybrid weighted averaging, we further develop intuitionistic fuzzy ordered weighted arithmetic averaging operator, intuitionistic fuzzy ordered weighted geometric averaging operator, intuitionistic fuzzy hybrid weighted arithmetic averaging (IFHWAA) operator, and intuitionistic fuzzy hybrid weighted geometric averaging (IFHWGA) operator. It is proved that all proposed weighted averaging operators have the properties of idempotency, boundary, monotonicity, and commutativity. Finally, we propose new methods based on IFHWAA and IFHWGA operators, respectively, to solve multi-attribute group decision making under intuitionistic fuzzy environment. Some examples are applied to illustrate the performance of the proposed methods. The experimental results show the effectiveness and advantages of the developed method by comparing with the other methods.

Generalized Ordered Weighted Reference Dependent Utility Aggregation Operators and Their Applications to Group Decision-Making

To quantify the influence of decision makers’ psychological factors on the group decision process, this paper develops a new class of aggregation operators based on reference-dependent utility functions (RUs) in multi-attribute group decision analysis. RUs include S-shaped RU and non-S-shaped RU. Each RU affords a framework where the psychological factors explicitly enter the decision problem via the basic utility function, reference point and loss aversion coefficient. Under the general framework, we derive a generalized ordered weighted S-shaped RU proportional averaging (GOSP) operator and a generalized ordered weighted non-S-shaped RU proportional averaging (GONSP) operator, respectively. The GOSP operator implies the risk attitude of the DM for relative losses is risk-seeking, while GONSP operator indicates the risk attitude in this case is risk-averse. As a special case, GONSP operator can degenerate into GOWPA operator which means that the attitude of the DM is risk-neutral. Each operator satisfies the desirable properties of general operator, i.e., monotonicity, commutativity, idempotency and boundedness. Furthermore, we consider hyperbolic absolute risk aversion (HARA) function as the basic utility function, and define an S-shaped HARA and a non-S-shaped HARA utility functions. Based on the two new RUs, we propose GOSP–HARA operator and GONSP–HARA operator. Every operator covers many existing aggregation operators. To ascertain weights of such operators, the paper builds an attribute-deviation weight model and a DMs-deviation weight model. Based on these RU operators and weight models, an approach is addressed for solving multiple attribute group decision-making problem. At last, an example is provided to show the feasible of our approach.

Linguistic multi-attribute decision making with multiple priorities

Linguistic multi-attribute decision making (MADM) problems have received many attentions in theoretical and practical aspects. It is well known that center of the linguistic MADM is the aggregation of fuzzy linguistic information. As one widely used type of aggregation operators, the importance weighted aggregation assumes that all the attributes are at the same priority level. This paper deals with linguistic MADM with multiple priorities in terms of strict priority hierarchy or weak priority hierarchy. To do so, based on the 2-tuple linguistic model and the revised product t-norm, a prioritized linguistic idempotent scoring (PLIS) operator is first proposed for linguistic MADM with a strict priority hierarchy to guarantee the idempotent property of prioritized aggregation. Moreover, as an extension of the PLIS operator, a PLIS ordered weighted averaging (PLIS-OWA) operator is proposed to solve linguistic MADM with a weak priority hierarchy. In addition, in practice decision-makers (DMs) are often unsure of their evaluations due to time pressure, lack of experience and data, and therefore may provide uncertain linguistic assessments. The PLIS and PLIS-OWA operators are also extended to the cases of prioritized MADM with interval-valued linguistic 2-tuples. Finally, three illustrative examples and comparative analysis are provided to show the effectiveness and efficiency of the proposed aggregation operators. The proposed operators outperform existing studies in terms of the idempotent property, no information loss, and heterogeneous information.

On multiplicative generators of overlap and grouping functions

Overlap and grouping functions have been proposed by Bustince et al.. In this paper, we continue investigating the two functions by their multiplicative generator pairs. At first, we introduce the concept of multiplicative generator pair for overlap functions. And then, we investigate the migrativity, homogeneity, idempotency, Archimedean and cancellation properties for the overlap functions obtained by such multiplicative generator pairs. Finally, we give the definition of multiplicative generator pair for grouping functions and show some related results by the duality of overlap and grouping functions.

Note on “Hesitant fuzzy prioritized operators and their application to multiple attribute decision making”

Motivated by the idea of prioritized aggregation (PA) operators, Wei (2012) developed two hesitant fuzzy prioritized aggregation (HFPA) operators, and discussed their desirable properties, but the definitions for the HFPA operators and their properties still need to be improved. In this short note, a numerical example is given to show that the idempotency of the HFPA operators suffers from certain shortcomings. Then, based on some adjusted operations on the hesitant fuzzy elements (HFEs), two improved aggregation operators are investigated to aggregate the collective of attribute values. We further prove that the improved operators have the properties of idempotency and boundedness. Finally, the comparison with the method proposed by Wei (2012) is performed to demonstrate that the proposed information aggregation method is both valid and practical to deal with decision making problems.

Area Efficient Sparse Modulo 2<sup>n</sup> - 3 Adder

This paper presents area efficient architecture of modulo 2n - 3 adder. Modulo adder is one of the main components for the implementation of residue number system (RNS) based applications. The proposed modulo 2n - 3 adder is implemented effectively, which utilizes parallel prefix and sparse concepts. The carries of some bits are calculated with the help of sparse approach in log2n prefix levels. This scheme is implemented with the help of idempotency property of the parallel prefix carry operator and its consistency. Parallel prefix structure contributes to fast carry computation. This will reduce area as well as routing complexity efficiently. The presented adder has double representation of residues in {0, 1, and 2}. The proposed adder offers significant reduction in area as the number of bits increases.

AN APPLICATION OF SOFT SET IN DECISION MAKING

In this paper, the definition of soft set and a detailed theoretical study of basic operations of soft sets such as intersection, extended intersection, restricted intersection, union, restricted union, complement and relative complement, Null and universal soft set are given. With the aid of definition of AND operation of soft sets and tabular representation of soft set, we are able to show that soft set has vital and real life application in decision making. The main aim of this paper is to use the concept of AND operation to sort out two best candidates out of five applicants in an interview conducted by a certain bank. Also the identification of Idempotent Property of “AND” and “OR” operation of soft sets is given and proved.

The Spectrum for $3$-Way $k$-Homogeneous Latin Trades

A $\mu$-way $k$-homogeneous Latin trade was defined by Bagheri Gh, Donovan, Mahmoodian (2012), where the existence of $3$-way $k$-homogeneous Latin trades was specifically investigated. We investigate the existence of a certain class of $\mu$-way $k$-homogeneous Latin trades with an idempotent like property. We present a number of constructions for $\mu$-way $k$-homogeneous Latin trades with this property, and show that these can be used to fill in the spectrum of $3$-way $k$-homogeneous Latin trades for all but $196$ possible exceptions.

On additive generators of overlap functions

In this paper we introduce the notion of additive generator pair for overlap functions. We also study how some of the properties that can be demanded to an overlap function can be expressed in terms of its generator pair. In particular, given a continuous and positive t-norm T, we analyze the overlap functions which are obtained by the distortion of T by a pseudo-automorphism, showing that the additive generator pair of such overlap function may be given in terms of the considered pseudo-automorphism and the additive generator of T. Finally, we also consider the influence of the migrativity, homogeneity and idempotency properties for the overlap functions obtained by such distortion in terms of their respective additive generator pairs.

Formalization and Verification of REST on HTTP Using CSP

Representational State Transfer (REST), as a promising software architecture style, has been used in large scale since proposed. But considerable confusions about REST exist and many examples of supposedly RESTful applications violate key REST constraints. In this paper, we focus on the most important constraints of REST, stateless property and hypertext-driven property. First we establish a formal model for REST on HTTP in CSP. In the model, components in RESTful systems communicate with each other using standard HTTP methods and are modeled as CSP processes. From the model we can find the effects of HTTP methods on resources. Then we give formal descriptions for failure cases of stateless, hypertext-driven constraints of REST, and safe, idempotent properties of HTTP methods, within which whether a system breaks REST constraints or basic HTTP requirements can be checked. Furthermore, we use model checker PAT to prove all the constraints hold in our model. In the end, a case study about the process of buying food is mapped to our model to better illustrate the REST concepts and our approach.

Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties

Overlap functions are a particular type of aggregation functions, given by increasing continuous commutative bivariate functions defined over the unit square, satisfying appropriate boundary conditions. Overlap functions are applied mainly in classification problems, image processing and in some problems of decision making based on some kind of fuzzy preference relations, in which the associativity property is not strongly required. Moreover, the class of overlap functions is reacher than the class of t-norms, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap functions. This flexibility of overlap functions increases their applicability. The aim of this papers is to introduce the concept of Archimedean overlap functions, presenting a study about the cancellation, idempotency and limiting properties, and providing a characterization of such class of functions. The concept of ordinal sum of overlap functions is also introduced, providing constructing/representing methods of certain classes of overlap functions related to idempotency, cancellation, limiting and Archimedean properties.

Weakly Prioritized Measure Aggregation in Prioritized Multicriteria Decision Making

This paper mainly investigates a special kind of multicriteria decision-making problem, in which all the criteria can be divided into several hierarchies and the criteria in the higher hierarchy have priorities over those in the lower hierarchy. It implies that the loss of the higher priority criterion can't be compensated by the gain of the lower prioritized criteria. As we know, fuzzy measures can well represent the interactions between criteria. In this situation, we develop a new fuzzy measure called weakly ordered prioritized measure WOPM to express the priority rule among the weakly ordered prioritized criteria. On the basis of the WOPM, we use discrete Choquet integral to construct a new WOPM-guided aggregation WOPMGA operator. To understand the priority property of this aggregation operator deeply, we get all the criteria's Shapley values and make an analysis of all criteria's Shapley values with different parameter values. Through analysis, we can find that the WOPMGA operator has the properties of boundedness, idempotency and monotonicity. Finally, we give several practical examples to illustrate the effectiveness of this aggregation operator.

Some properties of idempotents of (n, m)−semirings

In this paper, we investigate some properties of the additive and multiplicative idempotents of an (n, m)−semiring. In particular, the research is focused on differences between (n, m)−semirings, n, m ≥ 3 and usual semirings.