Lattice-valued preuniform convergence spaces

Using the intrinsic lattice-valued inclusion orders of L-subsets and of stratified L-filters, the concept of a stratified L-preuniform convergence structure is proposed. This convergence structure is asymmetric and can be used to establish a framework of asymmetric lattice-valued space structures. The category of stratified L-preuniform convergence spaces is introduced and some categorical properties are presented. A reflective subcategory of stratified L-preuniform convergence spaces is found that is categorically isomorphic to strong stratified L-convergence spaces (originally called stratified L-ordered convergence spaces). Several subcategories of stratified L-preuniform convergence spaces are established and relations between the different subcategories are presented. We conclude that stratified L-preuniform convergence structures introduced here could play a role in the framework of lattice-valued asymmetric space structures in lattice-valued topology.