L-pre-topology And Network Family Of Pre-convergence Class

In 1965,L.A.Zadeh estabilished the theory of fuzzy set.In 1968.C.L.Chang introduced the concept of fuzzy topological space,and some concepts in general topology were extended to fuzzy topological spaces.During the development of the theory of fuzzy topological spaces,the study on fuzzy pretopological spaces is proceeding.An L-topological space is an L-pretopological space,but not vice versa.With the deeper development of L-topology,L-pretopological spaces will play,a more important role in the study.Based on the work of the predecessors,the author continues to discuss the determination of L-pretopologies.For a given set X and a complete lattice L with the orderreversing involution',if we give an appropriate order relation(?) on CL(X)(the set of all L-preclosure operators) which makes CL(X) be a complete lattice that is ismorphic with(T(X),(?))(where T(X) is the set of all L-pretopologies on X).then we say that L-pretopologies on X can be determined by L-preclosure operators.In this paper,the author proves that an L-pretopology can be determined by an L-pre-R-neighborhood system operator,by an L-preexterior operator,or by an L-preboundary operator.The main points of this paper are as follows:In the first part of this paper,we define appropriate order relations on R(X)(the set of all L-pre-R-neighborhood system operators),E(X)(the set.of all L-preexterior operators) and B(X)(the set of all L-preboundary operators),respectively,and give ismorphisms(and their inverse mappings) from R(X),E(X).and B(X) to(T(X),(?)).The second part,is the continuation of the first,part.At frst.the notions of family of nets and preconvergence classes are introduced,then an injection from the set.of all pretopologies on a given set X to the set of all preconvergence classes of families of nets is given.Furthermore.it is proved that t,he injecti(?)n is a bijection if X is a finite set,. These results have improved the work of the predecessors.Although X.F.Yang has given the affirmative reply if L = {0.1}.the problem whether an L-preconvergence class on X determines an L-pretopology on X remains unresolved.