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In this paper, for a frameL, we characterize modified sobriety in stratifiedL-topological spaces and strongL-topological spaces internally using certain classes of stratifiedL-filters. While the first characterization using completely primeL-filters is trivial and applies to all stratifiedL-topological spaces, the other one using irreducibleL-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 strongL-topological spaces. [ABSTRACT FROM AUTHOR]