Research On Fuzzy Number Quotient Space

The theory of fuzzy numbers is an important part in fuzzy analysis. The quotient space of fuzzy numbers based on the equivalence relation determined by the symmetric fuzzy numbers is received a lot of attention. This thesis discusses the algebraic properties, convergence structure and uniform conver-gence of fuzzy function sequences in the quotient space of fuzzy numbers F/J. The main contests of this thesis are the following three aspects:Firstly, the addition of the equivalence classes of fuzzy number is intro-duced specifically in this thesis, and the relations among equivalence classes addition, midpoint function and Mares core are discussed. Then we study the connection between (F/J, +) and V[0,1], it is proved that (F/J, +) is isomorphic to V[0,1], which consists of the functions continuous from the right at 0, continuous from the left on (0,1], and the functions have bounded variation. In addition, we prove that (F/J,+,·) is a linear space in real number field.On the second part of this thesis, the properties of equivalence classes con-vergence in the sense of metric dsup are given, and compared with the conver-gence in in the sense of F/J, which is stronger.(F/J, +) is a topological group both in the sense of F/J and dsup. Moreover,(F/J,dsup) is a linear metric space. Then we introduce the concept and properties of level conver-gence, and give a necessary and sufficient condition for level convergence in F|J.The third part gives the concept of uniform convergence of fuzzy function sequences [a,b] ?F/J in the metric dsup, which could maintain the con-tinuity and integrability of uniform convergence fuzzy function sequences in F/J.