The granular partition lattice of an information table

In this paper we study the lattice of all indiscernibility partitions induced from attribute subsets of a knowledge representation system (information table in the finite case). This lattice, that we here call granular partition lattice, is a very well studied order structure in granular computing and data base theory and it provides a complete hierarchical classification of the knowledge obtained from all possible choices of attribute subsets. We show that it has a lattice structure also in the infinite case and we provide several isomorphic characterizations for this lattice. We discuss the potentiality of this order structure from both a micro-granular and a macro-granular perspective. Furthermore, the sub-poset of all the indiscernibility closures needed to determine when an arbitrary partition is an indiscernibility one is studied. Finally, we show the monotonic behaviour of the granular partition lattice with respect to entropy of partitions and attribute dependency in decision tables.