Algebraic Approach to Directed Rough Sets
This provides theoretical foundations for rough set theory with potential applications in formal concept analysis, student learning, and decision making.
The paper tackles the problem of developing algebraic semantics for directed rough sets by imposing minimal conditions on relations and using neighborhood granulations, proving necessary and sufficient conditions for complete distributivity in approximation lattices.
In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better behaved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approximations, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the relations, neighborhood granulations are used in the construction of all approximations (granular and pointwise). Necessary and sufficient conditions for the lattice of local upper approximations to be completely distributive are proved by the second author. These results are related to formal concept analysis. Applications to student centered learning and decision making are also outlined.