Characterization of stratified L-topological spaces by convergence of stratified L-filters

L-sets over a base set X are generalizations of classical sets where subsets are not specified by characteristic functions from X to {0, 1} but rather by functions from X to a lattice L. For an L-set a ∈ L and an element x ∈ X, a(x) is interpreted as the grade of membership of x in a. Stratified L-topological spaces are generalizations of topological spaces to the L-set case [1]. In [2], stratified L-generalized convergence spaces (analogous to classical convergence spaces) are defined, with the underlying lattice (L,≤,∧) being a frame. The resulting category SL-GCS is topological over Set and is Cartesian-closed [2]. SL-TOP, the category of stratified L-topological spaces, is isomorphic to a reflective subcategory of SL-GCS [2]. In [3] various subcategories of SL-GCS are investigated. The results of [2] and [3] are now extended to more general enriched lattices (L,≤, ∗,⊗). Finally axiom schemes for L-topological spaces based on L-filters (which lead to isomorphic categories in the frame case [4]) are investigated in the more general case and conditions for isomorphism between their categories are explored.