On supra R-open sets and some applications on topological spaces

In the present paper a new class of generalized supra open sets called supra R-open set is introduced. The relationships between some generalized supra open sets and this class are investigated and illustrated with enough examples. Also, new types of supra continuous maps, supra open maps, supra closed maps, and supra homeomorphism maps are studied depending on the concept of supra R-open sets. Finally, new separation axioms are dened and their several properties are studied

SUPRA HOMEOMORPHISM IN SUPRA TOPOLOGICAL ORDERED SPACES

The purpose of this paper is to introduce the concepts of x-supra continuous (open, closed, homeomorphism) maps in supra topological ordered spaces for x ∈ {I,D,B}. We study the relationship among these types with the help of examples and investigate the equivalent conditions for each concept. In particular, we present the sufficient conditions for maps to preserve some of separation axioms

A Comprehensive study on (α,β)-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation

The rough set (RS) and multi-granulation RS (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, the bipolar fuzzy sets (BFSs) are effective tools for handling bipolarity and fuzziness of the data. In this study, with the description of the background of risk decision-making problems in reality, we present $(\alpha, \beta)$-optimistic multi-granulation bipolar fuzzified preference rough sets ($(\alpha, \beta)^o$-MG-BFPRSs) and $(\alpha, \beta)$-pessimistic multi-granulation bipolar fuzzified preference rough sets ($(\alpha, \beta)^p$-MG-BFPRSs) using bipolar fuzzy preference relation (BFPR). Subsequently, the relevant properties and results of both $(\alpha, \beta)^o$-MG-BFPRSs and $(\alpha, \beta)^p$-MG-BFPRSs are investigated in detail. At the same time, a relationship among the $(\alpha, \beta)$-BFPRSs, $(\alpha, \beta)^o$-MG-BFPRSs and $(\alpha, \beta)^p$-MG-BFPRSs is given

Bipolar Soft Sets: Relations between Them and Ordinary Points and Their Applications

Bipolar soft set is formulated by two soft sets; one of them provides us the positive information and the other provides us the negative information. The philosophy of bipolarity is that human judgment is based on two sides, positive and negative, and we choose the one which is stronger. In this paper, we introduce novel belong and nonbelong relations between a bipolar soft set and an ordinary point. These relations are considered as one of the unique characteristics of bipolar soft sets which are somewhat expression of the degrees of membership and nonmembership of an element. We discuss essential properties and derive the sufficient conditions of some equivalence of these relations. We also define the concept of soft mappings between two classes of bipolar soft sets and study the behaviors of an ordinary point under these soft mappings with respect to all relations introduced herein. Then, we apply bipolar soft sets to build an optimal choice application. We give an algorithm of this application and show the method for implementing this algorithm by an illustrative example. In conclusion, it can be noted that the relations defined herein give another viewpoint to explore the concepts of bipolar soft topology, in particular, soft separation axioms and soft covers

(2,1)-Fuzzy sets: properties, weighted aggregated operators and their applications to multi-criteria decision-making methods

Abstract Orthopair fuzzy sets are fuzzy sets in which every element is represented by a pair of values in the unit interval, one of which refers to membership and the other refers to non-membership. The different types of orthopair fuzzy sets given in the literature are distinguished according to the proposed constrain for membership and non-membership grades. The aim of writing this manuscript is to familiarize a new class of orthopair fuzzy sets called “(2,1)-Fuzzy sets” which are good enough to control some real-life situations. We compare (2,1)-Fuzzy sets with IFSs and some of their celebrated extensions. Then, we put forward the fundamental set of operations for (2,1)-Fuzzy sets and investigate main properties. Also, we define score and accuracy functions which we apply to rank (2,1)-Fuzzy sets. Moreover, we reformulate aggregation operators to be used with (2,1)-Fuzzy sets. Finally, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making (MCDM) problems in the environment of (2,1)-Fuzzy sets. To show the effectiveness and usability of the proposed technique in MCDM problems, an illustrative example is provided

Generalized Frame for Orthopair Fuzzy Sets: (<i>m</i>,<i>n</i>)-Fuzzy Sets and Their Applications to Multi-Criteria Decision-Making Methods

Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “(m,n)-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank (m,n)-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with (m,n)-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of (m,n)-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example

Functionally Separation Axioms on General Topology

In this paper, we define a new family of separation axioms in the classical topology called functionally Ti spaces for i=0,1,2. With the assistant of illustrative examples, we reveal the relationships between them as well as their relationship with Ti spaces for i=0,1,2. We demonstrate that functionally Ti spaces are preserved under product spaces, and they are topological and hereditary properties. Moreover, we show that the class of each one of them represents a transitive relation and obtain some interesting results under some conditions such as discrete and Sierpinski spaces

Belong and Nonbelong Relations on Double-Framed Soft Sets and Their Applications

We aim through this paper to achieve two goals: first, we define some types of belong and nonbelong relations between ordinary points and double-framed soft sets. These relations are one of the distinguishing characteristics of double-framed soft sets and are somewhat expression of the degrees of membership and nonmembership. We explore their main properties and determine the conditions under which some of them are equivalent. Also, we introduce the concept of soft mappings between two classes of double-framed soft sets and investigate the relationship between an ordinary point and its image and preimage with respect to the different types of belong and nonbelong relations. By the notions presented herein, many concepts can be studied on double-framed soft topology such as soft separation axioms and cover properties. Second, we give an educational application of optimal choices using the idea of double-framed soft sets. We provide an algorithm of this application with an example to show how this algorithm is carried out

Novel Class of Ordered Separation Axioms using Limit Points

The paper aims to introduce a novel class of separation axioms on topological ordered spaces, namely Tci -ordered spaces (i = 0, 1 , 1, 1 1 , 2). They are defined by utilizing the notion of limit points of a set. With the aid of some examples, we scrutinize the 22 relationships between them as well as their relationships with strong Ti-ordered and Ti-spaces. Also, we investigate the interrelations between some of the initiated ordered separation axioms and some topological notions such as continuous topological ordered spaces and disconnected spaces. Furthermore, we verify that these ordered separation axioms are preserved under ordered embedding homeomorphism mappings and give a sufficient condition to be hereditary properties. Eventually, we demonstrate that the product of Tci -ordered spaces is also Tci -ordered for each i ̸= 2

Novel approaches of generalized rough approximation spaces inspired by maximal neighbourhoods and ideals

The theory of rough set is a robust approach to face uncertainty by classifying the data under study into three main regions. The main principles of this theory are approximation operators and accuracy measures, so improving them has been a major goal for several works. The intrinsic aim of this study is to create new methods with high accuracy measures to classify subsets of data. These methods are established by combining an ideal structure with four types of maximal neighbourhoods. The essential characterizations and properties of these methods are amply studied. Thereafter, the relationships between these methods are elucidated with the assistance of some numerical examples. In this regard, we prove that the approximation operators and accuracy measures induced from rough set model defined by intersection minimal-maximal neighborhoods are the best. In addition to that, some comparisons are provided to demonstrate the importance of the introduced techniques compared to the previous ones in terms of improving the approximation operators and increasing the values of accuracy. Finally, a numerical example is given to confirm the efficiency of the followed techniques to maximize the value of accuracy and shrink the boundary regions compared to the existing techniques