Multilattice as the set of truth values for fuzzy rough sets

Radzikowska and Kerre developed <i>L</i>-fuzzy rough set as a fuzzy generalisation of the notion of rough sets. Specifically, they have taken a residuated lattice <i>L</i> as the underlying set of truth degrees. However, in real life, we may encounter situations where truth degrees are not always linear, or where the existence of the least upper bound of two elements is no longer required. Instead, there may be the possibility of having minimal upper bounds, and dually, maximal lower bounds, multilattices have become a suitable algebraic structure for addressing such cases. Our aim is to extend Radzikowska and Kerre's work by replacing a residuated lattice with a multilattice <i>M</i>, thereby obtaining a more flexible structure. We define <i>M</i>-fuzzy rough sets using operators derived from adjoint pairs within the multilattice <i>M</i>. For a tolerance and an equivalence <i>M</i>-fuzzy relation, we define an appropriate <i>M</i>-fuzzy rough set and investigate its properties.