Structural analysis of indistinguishability operators and related concepts

Given an indistinguishability operator E, it is possible to obtain the set of extensional fuzzy subsets image and the respective operators of upper and lower approximations by extensional sets image and image. Reciprocally, given any of these objects, the initial indistinguishability operator E can be retrieved. It is well known that these concepts are in bijection. In this paper, we will prove that the relation underlying them is a lattice isomorphism. We will also consider further operators such as natural means and finally show the robustness of the results with respect to isomorphisms of t-norms.