Monoidal closedness of the category of T-semiuniform convergence spaces

Lattice-valued semiuniform convergence structures are important mathematical structures in the theory of lattice-valued topology. Choosing a complete residuated lattice L as the lattice background, we introduce a new type of lattice-valued filters using the tensor and implication operations on L, which is called T-filters. By means of T-filters, we propose the concept of T-semiuniform convergence structures as a new lattice-valued counterpart of semiuniform convergence structures. Different from the usual discussions on lattice-valued semiuniform convergence structures, we show that the category of T-semiuniform convergence spaces is a topological and monoidal closed category when L is a complete residuated lattice without any other requirements.