A Categorical Foundation of Rough Sets
Yoshihiko Kakutani
TL;DR
This work provides a category-theoretic foundation for rough sets by representing lower and upper approximations as adjoints to the data-induced functor $R^{*}$ between preordered sets. The approach yields a compositional theory of rough-set approximations, formalizes attribute reduction and data insertion, and enables automatic inference of decision rules for unseen data. It further extends the framework to refined lattices and enriched categories, unifying standard rough sets with fuzzy rough sets via complete residuated lattices and Kan extensions. Overall, the paper offers a scalable, principled basis for rough and fuzzy rough sets with potential applications to incomplete data analysis and advanced generalizations in enriched category theory.
Abstract
Rough sets are approximations of concrete sets. The theory of rough sets has been used widely for data-mining. While it is well-known that adjunctions are underlying in rough approximations, such adjunctions are not enough for characterization of rough sets. This paper provides a way to characterize rough sets in terms of category theory. We reformulate rough sets as adjunctions between preordered sets in a general way. Our formulation of rough sets can enjoy benefits of adjunctions and category theory. Especially, our characterization is closed under composition. We can also explain the notions of attribute reduction and data insertion in our theory. It is novel that our theory enables us to guess decision rules for unknown data. If we change the answer set, we can get a refinement of rough sets without any difficulty. Our refined rough sets lead rough fuzzy sets or more general approximations of functions. Moreover, our theory of rough sets can be generalized in the manner of enriched category theory. The derived enriched theory covers the usual theory of fuzzy rough sets.