Lattice-valued convergence spaces: Extending the lattice context

We define a category of stratified L-generalized convergence spaces for the case where the lattice is an enriched cl-premonoid. We then investigate some of its categorical properties and those of its subcategories, in particular the stratified L-principal convergence spaces and the stratified L-topological convergence spaces. For some results we need to introduce a new condition on the lattice (which is always true in the case where the lattice is a frame, but not always true in the more general case). As examples where we may apply the more general lattice context we examine the stratified L-topological spaces and probabilistic limit spaces. We show that the category of stratified L-topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a T-norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately.