Taking an algebraic perspective on the basic structures of Rough Concept
Analysis as the starting point, in this paper we introduce some varieties of
lattices expanded with normal modal operators which can be regarded as the
natural rough algebra counterparts of certain subclasses of rough formal
contexts, and introduce proper display calculi for the logics associated with
these varieties which are sound, complete, conservative and with uniform cut
elimination and subformula property. These calculi modularly extend the
multi-type calculi for rough algebras to a `nondistributive' (i.e. general
lattice-based) setting
