Diagonal conditions for lattice-valued uniform convergence spaces

We define diagonal conditions for lattice-valued uniform convergence spaces and show that these conditions are preserved under initial constructions. Further, the forgetful functor from the category of stratified lattice-valued uniform convergence spaces into the category of stratified lattice-valued limit spaces maps these conditions to corresponding conditions. We especially define uniform regularity and characterize this condition by certain closures of lattice-valued filters. As an application we generalize an extension theorem for uniformly continuous mappings.