Dualities and isomorphisms between indistinguishabilities and related concepts

Given a T-indistinguishability operator, the corresponding set H_E of extensional sets and the operators Φ_E and Ψ_E that provide upper and lower approximations of a fuzzy subset by extensional ones can be defined. Reciprocally, given any of these, the initial indistinguishability operator can be retrieved. In order to understand in depth how this relation works, it will be shown that there is an isomorphism between the corresponding lattices and how this isomorphism acts over the operations and orderings of the lattices will be studied.