Completely prime L-filters, irreducible L-filters and sobriety

Abstract

In this paper, for a frame L, we characterize modified sobriety in stratified L-topological spaces and strong L-topological spaces internally using certain classes of stratied L-filters. While the first characterization using completely prime L-filters is trivial and applies to all stratied L-topological spaces, the other one using irreducible L-filters generalizes an approach of R.E. Hoffmann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strong L-topological spaces.

Keywords: Lattice-valued filter, lattice-valued topology, sobriety