Defining rough sets using tolerances compatible with an equivalence

Abstract

We consider tolerances $T$ compatible with an equivalence $E$ on $U$, meaning that the relational product $E\circ T$ is included in $T$. We present the essential properties of $E$-compatible tolerances and study rough approximations defined by such $E$ and $T$. We consider rough set pairs $(X_E,X^T)$, where the lower approximation $X_E$ is defined as is customary in rough set theory, but $X^T$ allows more elements to be possibly in $X$ than $X^E$. Motivating examples of $E$-compatible tolerances are given, and the essential lattice-theoretical properties of the ordered set of rough sets $\{(X_E,X^T)\mid X\subseteq U\}$ are established.

In press, accepted manuscript