Double Successive Rough Set Approximations

TL;DR

This paper investigates double successive rough set approximations based on non-equivalent equivalence relations, exploring their reconstructability, uniqueness, and the conditions needed for their characterization on finite sets.

Contribution

It introduces conditions that characterize when pairs of equivalence relations generate unique double approximation operators and explores their reconstructability from these operators.

Findings

Characterization conditions for unique approximation operators

Reconstruction methods for equivalence relations from operators

Analysis of non-equivalent equivalence relations in rough set approximations

Abstract

We examine double successive approximations on a set, which we denote by $L_2L_1, \ U_2U_1, U_2L_1,$ $L_2U_1$ where $L_1, U_1$ and $L_2, U_2$ are based on generally non-equivalent equivalence relations $E_1$ and $E_2$ respectively, on a finite non-empty set $V.$ We consider the case of these operators being given fully defined on its powerset $\mathscr{P}(V).$ Then, we investigate if we can reconstruct the equivalence relations which they may be based on. Directly related to this, is the question of whether there are unique solutions for a given defined operator and the existence of conditions which may characterise this. We find and prove these characterising conditions that equivalence relation pairs should satisfy in order to generate unique such operators.