On complementation problem in the lattice of L-topologies

Abstract

In this paper, we study the lattice structure of the lattice FX of all L-topologies on a given nonempty set X. It is proved that the lattice FX is complemented and dually atomic when X is any nonempty set and membership lattice L is a complete atomic boolean lattice. Further we introduce the concept of limit point in the membership lattice and prove that if membership lattice L has a limit point, then for any nonempty set X, the lattice FX is not complemented.