Equivalence among L-closure (interior) operators, L-closure (interior) systems and L-enclosed (internal) relations

Closure (interior) operators and closure (interior) systems are important tools in many mathematical environments. Considering the logical sense of a complete residuated lattice L, this paper aims to present the concepts of L-closure (L-interior) operators and L-closure (L-interior) systems by means of infimums (supremums) of L-families of L-subsets and show their equivalence in a categorical sense. Also, two types of fuzzy relations between L-subsets corresponding to L-closure operators and L-interior operators are proposed, which are called L-enclosed relations and L-internal relations. It is shown that the resulting categories are isomorphic to that of L-closure spaces and L-interior spaces, respectively.