Object Similarity Measures and Pawlak's Indiscernibility on Decision Tables

In this paper we investigate the mathematical foundations of the notion of similarity between objects in relation to the granulations on a decision table D. First of all, we compare the endogenous granulation induced by Pawlak’s indiscernibility with the exogenous granulation induced by a similarity measure ζ defined on pairs of objects and assuming values in the unit interval. To this aim, the starting point of our analysis is the introduction of the notion of refinement of the granulation induced by an attribute subset A through the object similarity measure ζ. More in detail, we say that ζ refines the granulation induced by A if ζ assumes value 1 on a pair of objects if and only if they are A-indiscernible. Next, starting from two given families ρ and ν of numerical maps defined on pairs of admissible values of D, we determine a broad class of potential similarity measures on the objects of D refining, sometimes under some specific additional hypotheses, the A-granulation on the object set of D. With regard to a such class of similarity measures, we establish several mathematical properties. Finally, we focus our attention to the analysis of specific pairs of numerical maps ρ and ν that have been classically studied in literature and, for each of them, we exhibit the main properties with respect to the aforementioned refinement of granulation.