Multiset Theory Research Articles

Theoretical aspects of designing the components of an aircraft onboard cable network

The article presents theoretical foundations of designing the components of an aircraft onboard cable network, proposed as a methodology for solving the problems of designing onboard equipment. The composition of the initial and developed technical documentation for the aircraft onboard cable network is determined. The article also shows the relationship between the stages of designing the onboard cable network and the stages of designing the aircraft. The formulation of the general task of creating the components of an onboard cable network is presented as a number of particular subtasks of a lower level of complexity. The method for solving the problem is based on a number of transformations performed with the original documentation in order to develop technical documentation for the introduction of the onboard system into the complex of onboard equipment. A number of successive transformations of the proposed bundle model using the apparatus of topology and multiset theory made it possible to form a model of the components of the onboard cable network of an aircraft that gives a detailed view of its future design. The proposed scientific approach to the design of aircraft onboard cable network components created the prerequisites for minimizing the heuristic component in the process of designing aircraft onboard cable network components.

Axiomatization multisets: a comparative analysis

A multiset, unlike the classical set, allows for multiple instances of its elements. In this paper, we present a comparative analysis of theories on multisets. In particular, we examine, the first-order two-sorted multiset theory MST, and the single-sorted multiset theory MS that employs the same sort for multiplicities and the set they support. The logical strengths and significance of some axioms presented in these theories are investigated. The theory MST contains a copy of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) but is independent of ZFC. The single-sorted multiset theory describes a stronger theory that mirrors the Zermelo-Fraenkel set theory (ZF) and is equiconsistent with ZF and antifoundation. The two-sorted multisettheory MST is a conservative extension of the classical set theory, making it a suitable theory to assume when dealing with studies that involve multisets.

METRICS AND METRIC SPACES OF SOFT MULTISETS

The theory of Soft set found applications in so many fields including multiset theory to obtain soft multisets. These theories together with some of their properties were presented. Moreover, considering the various applications of metric spaces in various fields; Metrics and metric spaces of soft multisets with some of their attributes were introduced. However, it was discovered that only pseudo-metric spaces could favorably be formulated. Moreover, soft multiset ordering was also presented.

An approach to solution of constrained clustering problems using the constraint programming paradigm and the multiset theory

An integrated approach is proposed to solve the Constrained Clustering problems which are oriented on taking the opinion of a group of experts into account. A cluster and the objects to be clustered are proposed to be represented as multisets, and the distance between these to be determined by metrics in the Petrovsky multisets space. The approach is implemented within the Constraint Programming paradigm. In so doing, the significant complexity is in organizing effective processing the qualitative constraints, namely, the rules for assigning the objects to one or different clusters. The qualitative constraints are proposed to be represented and processed as table constraints of a new type, i.e., as the smart-tables of D-type. The main attention is confined to the problem of how to reduce the amount of constraints and how to simplify the Constrained Clustering problem. It is proposed to generate constraints for some pairs of objects rather than for all of them, with the generation based on a priori interval estimation of the optimal value of the clustering criterion. To do that, a modified method of multisets hierarchical clustering has been proposed. The approach proposed allows a global optimum to be found for the Constrained Clustering problems considered.

Linguistic approach to the classification problem based on the multiset theory

The problem of developing generalizing decision rules for object classification, which arises under conditions of inaccurate knowledge about the values of objects’ attributes, and about the significance of the attributes themselves, has been considered. The approach to the binary classification of objects, which implements the representation of inaccurate knowledge based on linguistic variables and allows one to consider various strategies for the formation of generalizing decision rules for classification using the tools of multiset theory, has been proposed. The example of the formation of generalizing decision rules of binary classification for the set of competitive projects evaluated by the group of experts, confirming the effectiveness of the proposed approach, has been considered. A herewith, visualization of objects, the values of the features of which are the frequency of setting a certain score according to the a priori given rating scale by all experts, in a two-dimensional space using the non-linear dimensionality reduction algorithm named as the UMAP algorithm, has been implemented. Based on the results of visualization and cluster analysis of the initial set of competitive projects, the “noise” project, which negatively affects the results of the formation of generalizing decision rules of binary classification, was identified and removed from further analysis.

Neutrosophic Multigroup Homomorphism and Some of its Properties

In a way, the notion of neutrosophic multigroup is an application of neutrosophic multisets to the theory of group. The concept of neutrosophic multigroup is an algebraic structure of neutrosophic multiset that generalizes both the theories of classical group and neutrosophic group. Neutrosophic multigroup constitutes an application of neutrosophic multiset to the elementary theory of classical group. In this paper, we propose the concept of homomorphism on neutrosophic multigroup. We define homomorphism kerlf, automorphism, homomorphic image and homomorphic preimage of neutrosophic multigroup, respectively. Some homomorphic properties of neutrosophic multigroup are explicated. Some homomorphic properties of neutrosophic multigroup are also discussed. This new concept of homomorphism as a bridge among set theory, fuzzy set theory, neutrosophic multiset theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroup homomorphism and give its applications to group theory

Методы групповой классификации на основе теории мультимножеств в задаче локализации зон с различным уровнем сейсмической активности при ведении горных рабо

The work is dedicated to assessing the applicability of supervised group classification methods developed on the basis of multiset theory for solving the problem of identifying zones with different degrees of seismic activity (using the example of one of the sections of the highly stressed rock massif of the Kukisvumchorr apatite-nepheline deposit). The initial objects for classification procedures are spatial cells into which the fieldis divided. Each spatial cell is described by a certain set of factors that, according to experts, have an impact on the occurrence of seismic events in a given cell. An original representation of spatial cells (their groups) as a set of multisets is proposed. Studies have been carried out aimed at identifying the influence of various options for presenting the initial data on the result of classification procedures. Representation of objects described by quantitative and / or qualitative features and existing in several versions (copies) in the form of multisets makes it possible not to transform qualitative features into numerical ones when performing clustering procedures and use methods of group classification of objects. Generalized decision rules of group classification for assigning objects (spatial cells) to four classes of seismic hazard are obtained. In contrast to the currently widely used technologies based on the neural network approach, in this work, the training result is not a “black box” in the form of a trained neural network, but a set of rules that can be easily interpreted, which increases the confidence of end users in decision-making procedures.

Soft multi-rough set topology with applications to multi-criteria decision-making problems

Rough set theory introduced by Pawlak (Int J Comput Inf Sci 11:341–356, 1982), multi-set theory proposed by Blizard (Notre Dame J Form Log 30:36–65, 1989) and soft set theory introduced by Molodtsov (Comput Math Appl 37(4–5):19–31, 1999) are fundamental concepts in computational intelligence, which have a myriad of applications in modeling uncertainties and decision making under uncertainty. In this paper, the idea of soft multi-rough set (SMRS) is introduced as a hybrid model of soft set, multi-set and rough set. The SMRS provides roughness of a multi-set in terms of soft multi-approximation space. The novel concept of soft multi-rough topology (SMR-topology) is defined to discuss topological structure of SMRSs by using pairwise SMR-approximations. The proposed models of SMRS and SMR-topology are suitable for modeling uncertainties in the real-life circumstances. SMR-topology is the generalization of crisp topology, soft topology and soft rough set topology. Some fundamental properties of SMR-topology and their related results are studied. Some algorithms for are developed for multi-criteria decision making based on soft multi-sets, soft multi-rough sets and soft multi-rough topology. Based on proposed algorithms, the applications of SMRSs and SMR-topology toward diagnosis of depression and diabetes are illustrated by the numerical examples. A comparison analysis of proposed methods with some existing methods is also given to justify their reliability, feasibility and flexibility.

Rule-Based Classification using Multi Soft Set Theory

One of the important data mining functionality is classification. Presently, different methods exist for implementing classification. Rule-based classification using decision tree induction method is a conventional and simple method for identifying an unknown class of a given object. This method has a set of demerits and to remove these demerits, we depend on a soft computing tool which is known as soft set theory. One branch of soft set theory is called - multi soft theory- and it has a wide range of applications in the area of classification. We made a certain alteration in the rule-based classification using decision tree induction method by applying multi soft set theory. These changes will simplify the difficulties of the rule-based classification using decision tree induction method. The first two sections of this research work discuss introduction and preliminaries. In the remaining sections, the authors describe the multi soft set theory and its applications in rule base classification. Lastly, the paper finishes with a new algorithm, which the research scholars implemented as software using python programming. The suggested work experts can use in data mining industry. It has massive use in the fields of business, agriculture, health, education and many more.

Multisets, heaps, bags, families: What is a multiset?

AbstractIs the current formulation of multiset theory, which is based on sets and multiplicities of their elements, adequate? We exhibit both mathematical and metamathematical reasons which should cause one to rethink the definition. Some problems with multiset theory in its accepted formulation concern even the basic operations of union, intersection, and complement; others, more deeply rooted, concern Cartesian products, relations, or morphisms. We compare current definitions and conclude that the problems of multiset theory need to be resolved at the fundamental level of sets and mappings (or equivalent constructs) with multiplicities introduced only as a secondary concept. As a consequence, we propose to define multisets as families. A mapping establishes the connection to the familiar theory of multisets. Without losing anything, our proposal is simple and provides for an elegant mathematical theory.

Selecting a Single Result from an Aggregate of Contradictory Alternatives with Use of Multiset Theory

We consider basic approaches to decision-making using computer systems. Ambiguous results in decision-making are obtained through the use of different methods for source-data processing. In order to obtain a single result from an aggregate of such decisions, it is suggested to apply the principle of selecting the decision that is most proposed by John von Neumann to increase the reliability of computers. Based on multiset theory, we develop a mathematical model of selecting a single result from an aggregate of alternatives obtained using different methods of decision-making. In this model, a decision is selected based on computing the argument of maximizing the multiplicity functions of elements for the arithmetic sum of multisets of the obtained decisions. The peculiarities of applying this approach are considered and methods of adjusting the source data for obtaining a single result are proposed. By the example of selecting a metal for electroplating products we present the results (of the decisions taken) obtained by a group of experts using the described approach. Recommendations for improving the efficiency of the proposed approach are presented.

Neutrosophic Soft Multi-Set Theory and Its Decision Making

In this study, we introduce the concept of neu- trosophic soft multi-set theory and study their properties and operations. Then, we give a decision making meth- ods for neutrosophic soft multi-set theory. Finally, an ap- plication of this method in decision making problems is presented.

Neutrosophic Multigroups and Applications

In recent years, fuzzy multisets and neutrosophic sets have become a subject of great interest for researchers and have been widely applied to algebraic structures include groups, rings, fields and lattices. Neutrosophic multiset is a generalization of multisets and neutrosophic sets. In this paper, we proposed a algebraic structure on neutrosophic multisets is called neutrosophic multigroups which allow the truth-membership, indeterminacy-membership and falsity-membership sequence have a set of real values between zero and one. This new notation of group as a bridge among neutrosophic multiset theory, set theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroups and give its applications to group theory.

Synopsis of the Notions of Multisets and Fuzzy Multisets

This paper is an attempt to summarize the basic concepts of the theories of multiset and fuzzy multiset. We begin by describing multisets and the operations between them with some related results. In the same vein, the basic concepts of fuzzy multiset theory as well as the operations between fuzzy multisets are buttressed in relation to multiset theory. Finally, we present some properties of fuzzy multisets with some related results.

A study of hesitant fuzzy soft multiset theory

A study of hesitant fuzzy soft multiset theory

Application of multiset theory for the selection of the single result from alternatives aggregate obtained using different decision methods

In this article are discussed the main reasons to obtain different results in decision-making by various methods. It is proposed to use the multiset theory for selection a single result from a solution aggregate. In the work is described mathematical decision-making model based on the argument calculating for maximizing the elements multiplicity functions of the arithmetic sum to found multiset solutions. Here are presented decisions compare results, which were made by expert group and according to the approach proposed by the authors on an example of the galvanic coating choice. In this article are represented recommendations for improving the efficiency of the proposed approach for obtaining the unit dimension of the resulting multiset solutions.

A Single-Sorted Theory of Multisets

An axiomatic account of multiset theory is given, where multiplicities are of the same sort as sets (with their own internal structures). Various theories are proposed covering different existing multiset systems, as well as a stronger theory which is equiconsistent with Zermelo–Fraenkel set theory and with antifoundation. The inclusion relation receives a recursive definition in terms of membership and is shown to be not always antisymmetric.

Binary relations and algebras on multisets

Contrary to the notion of a set or a tuple, a multiset is an unordered collection of elements which do not need to be different. As multisets are already widely used in combinatorics and computer science, the aim of this paper is to get on track to algebraic multiset theory. We consider generalizations of known results that hold for equivalence and order relations on sets and get several properties that are specific to multisets. Furthermore, we exemplify the novelty that brings this concept by showing that multisets are suitable to represent partial orders. Finally, after introducing the notion of an algebra on multisets, we prove that two algebras on multisets, whose root algebras are isomorphic, are in general not isomorphic.

Topological approaches to generalized definitions of rough multiset approximations

This paper proposes new definitions of lower and upper multiset approximations, which are basic concepts of the rough multiset theory. These definitions come naturally from the concepts of multiset topology and the concepts of ambiguity introduced in this paper. The new definitions are compared to the classical definitions and are shown to be more general. In the sense that they are the only ones which can be used for any type of indiscernibility or similarity multiset relation.

ENCODINGS OF MULTISETS

The purpose of this paper is to present various algebraic views of multisets, and certain connections between the theory of multisets (with multiplicities in the semiring of positive integers) and natural computing, in particular membrane and DNA computing. We introduce a Gödel encoding of multisets, and find some results regarding this encoding together with new connections and interpretations. We also introduce the norm of a multiset and we find some relationships between multiset theory and number theory.