Rough sets determined by tolerances

• We consider rough set systems RS determined by a tolerance relation R.
• We give a necessary and sufficient condition for RS to be a complete lattice.
• RS is a completely distributive lattice iff R is induced by an irredundant covering.
• We introduce the formal concept representation of rough sets as a completion of RS.
• Motivating examples related to information systems are presented.

We show that for any tolerance R on U, the ordered sets of lower and upper rough approximations determined by R form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if R is induced by an irredundant covering of U, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set RS of rough sets determined by a tolerance R on U is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that R is a tolerance induced by an irredundant covering of U if and only if RS is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on RS. We present necessary and sufficient conditions which guarantee that for a tolerance R on U  , the ordered set RSXRSX is a lattice for all X⊆UX⊆U, where RXRX denotes the restriction of R to the set X   and RSXRSX is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are Dedekind–MacNeille completions of RS.