Micro and Macro Models of Granular Computing induced by the Indiscernibility Relation

In rough set theory (RST), and more generally in granular computing on information tables (GRC-IT), a central tool is the Pawlak’s indiscernibility relation between objects of a universe set with respect to a fixed attribute subset. Let us observe that Pawlak’s relation induces in a natural way an equivalence relation ≈ on the attribute power set that identifies two attribute subsets yielding the same indiscernibility partition. We call indistinguishability relation of a given information table I the equivalence relation ≈, that can be considered as a kind of global indiscernibility. In this paper we investigate the mathematical foundations of indistinguishability relation through the introduction of two new structures that are, respectively, a complete lattice and an abstract simplicial complex. We show that these structures can be studied at both a micro granular and a macro granular level and that are naturally related to the core and the reducts of I. We first discuss the role of these structures in GrC-IT by providing some interpretations, then we prove several mathematical results concerning the fundamental properties of such structures.