Connections of L-Quasi-Internal and L-Quasi-Enclosed Spaces with L-Grills, L-Filters and L-Neighborhood Spaces

The paper establishes a categorical framework linking L-quasi-internal and L-quasi-enclosed spaces with L-filter, L-grill, and L-neighborhood spaces within the context of complete lattices. We define and characterize L-quasi-internal and L-quasi-enclosed relations, demonstrating how these structures can be mutually induced by L-filter and L-grill spaces. Specifically, the study proves the existence of categorical Galois connections between the category of L-grill spaces and L-quasi-enclosed spaces, along with those between L-filter spaces and L-quasi-internal spaces. Additionally, we investigate the properties of stratified and strong L-quasi-structures, clarifying their interrelationships with L-neighborhood systems. The theoretical results are reinforced by illustrative examples and a practical application in epidemiological risk assessment, demonstrating the usefulness of the proposed framework for decision-making under uncertainty.