Some Results Related to the Lattice of Fuzzy Topologies on a Fixed Set

Abstract

The set of all L - fuzzy topologies on a fixed set X is a complete lattice denoted by LFT(X,L). In this paper, we determine some classes of automorphisms of this lattice when X is a nonempty set and L is an F- lattice. In 1958, Juris Hartmanis (2) determined the automorphisms of the lattice LT(X) of all topologies on a fixed set X as follows : for p  S(X) and   LT(X), define the mapping Ap by Ap() = { p(U) : U }. Then Ap() is a topology on X and Ap is an automorphism of LT(X). If X is infinite or X contains atmost two elements, the set of all automorphisms of LT(X) is precisely {Ap : p  S(X)}. Otherwise, the set of all automorphisms of LT(X) is {Ap : p S(X)}  {Bp : p S(X)} where Bp : LT(X) LT(X) is defined by Bp() = { X-p(U): U} for  LT(X). From this result, we can conclude that, if X is an infinite set and P is any topological property, then the set of topologies in LT (X) possessing the property P may be identified simply from the lattice structure of LT(X), since the only automorphisms of LT(X) for infinite X are those which simply permute elements of X. Therefore any automorphism of LT(X) must map all the topologies in LT(X) onto their homeomorphic images. Thus the topological properties of elements of LT(X) must be determined by the position of the topologies in LT(X). In this paper, we determine some classes of automorphisms of lattice LFT(X,L) where L is a complete, distributive and pseudo complemented lattice (or an F - lattice).