Some Topological and Algebraic Properties of α-level Subsets’ Topology of a Fuzzy Subset

Abstract

The theory of fuzzy sets, since its foundation, has advanced in a wide range of means and in many fields. One of the areas to which fuzzy set theory has been applied extensively is mathematical programming. Nevertheless, the applications of fuzzy theory can be found in e.g. logic, decision theory, artificial intelligence, computer science, control engineering, expert systems, management science, operations research, robotics, and others. Theoretical improvements have been made in many directions. Nowadays it has a lot of applications also on possibility theory, actuarial credibility theory, fuzzy logic and approximate reasoning, fuzzy control, fuzzy data analysis, fuzzy set models in operations research, etc. The aim of this paper is to investigate some topological properties of a set X when the topology defined on it is the collection of all the α-level subsets of a fuzzy subset A of X.We have been able to establish some results regarding fuzzy cluster level subsets, convergence of level subsets and quasicompactness among others.

On Single-Valued Neutrosophic Ideals in Šostak Sense

“…He [20] further evidenced that smooth topologies collection constitutes a complete lattice. Furthermore, Onassanya and Hošková-Mayerová [21] inspected certain features of subsets of α-level as an integral part of a fuzzy subset topology. Likewise, more specialists in the field like Çoker and Demirci [22], in addition to Samanta and Mondal [23,24], have provided definitions to the concept of graduation intuitionistic openness of fuzzy sets based on Šostak's sense [25] according to Atanassov's [26] intuitionistic fuzzy sets.…”

Ordinary Single Valued Neutrosophic Topological Spaces

“…Al Tahan et al [21] defined a topology such that the hyperoperation is pseudocontinuous, and showed that there is no relation in general between pseudotopological and strongly pseudotopological hypergroupoids. In addition, Onassanya and Hošková-Mayerová [22] investigated some topological properties of α-level subsets' topology of a fuzzy subset. Moreover, Çoker and Demirci [23], and Samanta and Mondal [24,25] defined intuitionistic gradation of openness (in short IGO) of fuzzy sets inŠostak's sense [26] by using intuitionistic fuzzy sets introduced by Atanassov [27].…”

The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

“…Years later, Demirci [4] defined "smooth groups" by using fuzzy functions and fuzzy equalities [5][6][7]. Since then, many researchers have worked in various directions, such as with smooth subgroups and smooth homomorphisms, generalized vague groups, vague rings, vague ideals, fuzzy multi-polygroups, fuzzy multi-H v -ideals, and other fuzzy algebra [8][9][10][11][12][13][14][15][16][17][18][19].Liu [20,21] proposed and obtained fundamental results of fuzzy subrings and fuzzy ideals of a ring. Dixit et al [22] discussed some basic properties of fuzzy rings.…”