Lattice-valued semiuniform convergence spaces

In this paper, two kinds of lattice-valued semiuniform convergence spaces are proposed, namely stratified L-semiuniform convergence spaces and stratified L-ordered semiuniform convergence spaces respectively. It is shown that (i) the category of stratified L-semiuniform convergence spaces is topological; (ii) the category of stratified L-ordered semiuniform convergence spaces is a bireflective full subcategory of the category of stratified L-semiuniform convergence spaces, and hence it is topological; (iii) both the category of stratified L-semiuniform convergence spaces and that of stratified L-ordered semiuniform convergence spaces are Cartesian-closed; (iv) the category of stratified L-semiuniform convergence spaces is extensional; (v) both the category of stratified L-semiuniform convergence spaces and that of stratified L-ordered semiuniform convergence spaces are closed under the formation of products of quotient mappings. In case that L is the two-point chain, both coincide with the category of semiuniform convergence spaces in the classical case.