The Research On The Theory Of (L,M)-fuzzy Convergence

This thesis mainly contains three parts. The first part contains Chapter 2, Chapter 3 and Chapter 4. In this part, the theories of(L, M)-fuzzy Q-convergence structures,(L, M)-fuzzy strong convergence structures and(L, M)-fuzzy convergence structures are established and their relations with(L, M)-fuzzy topologies are studied, respectively. The second part is Chapter 3, the aim of this part is to establish the compactness theory in(L, M)-fuzzy convergence spaces and apply this theory to define the compactness degrees of(L, M)-fuzzy topological spaces,(L, M)-fuzzy pseudo-quasi-metric spaces and pointwise(L, M)-fuzzy quasi-uniform spaces. The third part is Chapter 6, in this part, the concept of pointwise pseudo-metric chains is introduced and its relations with(L, M)-fuzzy pseudometrics,(L, M)-fuzzy topologies and pointwise(L, M)-fuzzy uniformities are investigated. In the following, let me explain explicitly what I have done.Chapter 1 is introduction. In this chapter, a general survey of this thesis as well as preliminaries on category theory, lattice theory and(L, M)-fuzzy topologies are given, which will be used throughout this thesis.In Chapter 2, the theory of(L, M)-fuzzy Q-convergence spaces is established. Firstly,the concept of(L, M)-fuzzy Q-convergence structures is defined. It is shown that there is an adjunction between the category of(L, M)-fuzzy Q-convergence spaces and the category of(L, M)-fuzzy topological spaces. In particular, the category of(L, M)-fuzzy topological spaces can be embedded in the category of(L, M)-fuzzy Q-convergence spaces as a reflective full subcategory. Secondly, the notions of pretopological(L, M)-fuzzy Q-convergence structures and topological(L, M)-fuzzy Q-convergence structures are defined and their characterization theorems are presented. It is also proved that the category of topological(L, M)-fuzzy Qconvergence spaces is a bireflective subcategory of the category of pretopological(L, M)-fuzzy Q-convergence spaces, and the former is isomorphic to the category of(L, M)-fuzzy topological spaces. Finally, several separation axioms are defined in(L, M)-fuzzy Q-convergence spaces, and the productivity with respect to these separation properties are discussed.In Chapter 3, the theory of(L, M)-fuzzy strong convergence spaces are established.Firstly, two kinds of lattice-valued convergence structures are defined, which are called stratified(L, M)-fuzzy strong convergence structures and(L, M)-fuzzy strong convergence structures, respectively. It is proved that there is an adjunction between the category of(stratified)(L, M)-fuzzy strong convergence spaces and the category of(stratified)(L, M)-fuzzy topological spaces. Specially, the latter can be embedded in the former as a reflective full subcategory. Secondly, it is shown that the category of stratified(L, M)-fuzzy strong convergence spaces is a Cartesian-closed topological category. Finally, a class of special kinds of(stratified)(L, M)-fuzzy strong convergence spaces are introduced and their categorical properties are investigated.In Chapter 4, the theory of(L, M)-fuzzy convergence spaces is established. Firstly,the concept of(L, M)-fuzzy convergence spaces is introduced and it is proved that there is an adjunction between the category of(L, M)-fuzzy convergence spaces and the category of(L, M)-fuzzy topological spaces. In particular, the latter can be embedded in the former as a reflective full subcategory and the category of(L, M)-fuzzy convergence spaces is a Cartesian-closed topological category. Secondly, pretopological and topological(L, M)-fuzzy convergence spaces are proposed. It is shown that the category of topological(L, M)-fuzzy convergence spaces is isomorphic to the category of(L, M)-fuzzy topological spaces and they are both bireflective subcategories of the category of pretopological(L, M)-fuzzy convergence spaces. Also, the category of pretopological(L, M)-fuzzy convergence spaces is a bireflective subcategory of the category of(L, M)-fuzzy convergence spaces. Finally, the relations among(L, M)-fuzzy Q-convergence spaces,(L, M)-fuzzy strong convergence spaces and(L, M)-fuzzy convergence spaces are investigated. It is proved that the category of(L, M)-fuzzy Q-convergence spaces can be embedded in the category of(L, M)-fuzzy strong convergence spaces as a reflective full subcategory and the latter can be embedded in the category of(L, M)-fuzzy convergence spaces as a coreflective full subcategory.– VI –In Chapter 5, the theory of compactness of(L, M)-fuzzy convergence spaces is established. Firstly, the degrees of compactness of(L, M)-fuzzy convergence spaces are defined by using the convergence of(L, M)-fuzzy ultrafilters. Also, the lattice-valued Tychono? Theorem is presented. Secondly, based on the convergence of(L, M)-fuzzy ultrafilters, a new definition of degrees of compactness of(L, M)-fuzzy topological spaces is introduced and the latticevalued Tychono? Theorem in the framework of(L, M)-fuzzy topological spaces is presented.Finally, it is shown that an(L, M)-fuzzy pseudo-quasi-metric can induce a pointwise(L, M)-fuzzy quasi-uniformity. The degrees of compactness of(L, M)-fuzzy pseudo-quasi-metric spaces and pointwise(L, M)-fuzzy quasi-uniform spaces are defined by using(L, M)-fuzzy convergence structures.In Chapter 6, the theory of pointwise pseudo-metric chains is established. In this chapter, the relations between(L, M)-fuzzy pseudo-metrics and pointwise pseudo-metric chains are discussed. It is shown that there is a one-to-one correspondence between(L, M)-fuzzy pseudo-metrics and pointwise pseudo-metric chains which preserve- operations. Secondly, it is shown that a pointwise pseudo-metric chain can induce an(L, M)-fuzzy topology and this topology coincides with the topology generated by its induced(L, M)-fuzzy pseudometric. Finally, it is also proved that a pointwise pseudo-metric chain can induce a pointwise(L, M)-fuzzy uniformity and this uniformity coincides with the uniformity generated by its induced(L, M)-fuzzy pseudo-metric.Chapter 7 is conclusion. In this chapter, the main contribution of this thesis is summarized and the expectation are made.