Categorical relationships between $L$-grill-proximity spaces and related $L$-topological structures

This work investigated categorical connections between $L$-grill spaces and $L$-pre-proximity spaces and established a Galois connection between the category of $L$-pre-proximity spaces and the category of stratified $L$-grill spaces within the framework of complete residuated lattices. Furthermore, it introduced the concept of $L$-grill-proximity spaces, a new structure derived from $L$-grills, and studied their fundamental properties as well as their categorical connections with $L$-closure, $L$-interior, and $L$-co-topological spaces. Finally, it demonstrated the applicability of the proposed framework through an application to environmental pollution modeling by integrating multiple environmental indicators for quantitative air-quality assessment and decision-making.