  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.
