The rough powerset monad
Rough sets and fuzzy sets are both methods to represent uncertainty. In previous work we have developed, within the abstract language of category theory, some interesting tools for providing a foundation to the development of a general framework for unification, working with powersets of terms. Monads, in this context, establish an essential concept in that they contain set functors with structure provided by natural transformations. In this paper we show how monads, extended to partially ordered monads, can be used to generalize and interpret rough situations. In particular, the partially ordered ordinary power set monad turns out to contain sufficient structure in order to provide rough set operations. This study of rough sets from a categorical view, provides an abstract tool to handle properties of the structure increasing their understanding in a basic many-valued logic setting.
monads rough sets category theory partially ordered monads many-valued logic similarity relations