Granular Computing on Information Tables: Families of Subsets and Operators.

In this work we use the granular computing paradigm to study specific types of families of subsets, operators and families of ordered pairs of sets of attributes which are naturally induced by information tables. In an unifying perspective, by means of some representation results, we connect the study of finite closure systems, matroids and finite lattice theory in the scope of the more general notion of attribute dependency based on information tables . For a fixed finite set $Omega$ and for a corresponding information table $mcI$ having attribute set $Omega$, the fundamental tool we use to proceed in our investigation is the equivalence relation $approxmi$ on the power set $mcP(Omega)$ which identifies two any sets of attributes inducing the same indiscernibility relation on the object set of $mcI$. We interpret the attribute dependency as a preorder $gemi$ on $mcP(Omega)$ whose induced equivalence relation coincides with $approxmi$. Then we investigate in detail the links between the preorder $gemi$, a closure system and an abstract simplicial complex on $Omega$ naturally induced by $approxmi$ and specific families of ordered pairs of sets of attributes on $Omega$.