The category of residuated lattice valued filter spaces

Considering L as a complete residuated lattice, some further investigations on L-filter spaces are made. Firstly, it is shown that the category L-Fil of L-filter spaces is monoidal closed. Secondly, it is proved that the categories of L-Cauchy spaces and L-semi-Cauchy spaces are both bireflective subcategories of L-Fil. Finally, the concept of filter-determined L-semiuniform convergence spaces is proposed and the resulting category is shown to be isomorphic to L-Fil, which can be embedded in the category of L-semiuniform convergence spaces as a bicoreflective subcategory. Moreover, it is proved that L-Fil can be embedded in the category of L-semiuniform convergence spaces as a bireflective subcategory whenever L is a completely distributive lattice.