The topological structure in parameterized rough sets

In this paper, by strict mathematical reasoning, we discover the relation between the parameters and the reducts in parameterized rough reduction. This relation, named the nested reduction, shows that the reducts can act as a nested structure with the monotonically increasing parameter. Furthermore, we present a systematic theoretical framework which gives some basic principles to construct the nested structure in parameterized rough reduction. What is more, some specific parameterized rough set models, in which the nested reduction can be constructed, are pointed out by strict mathematics reasoning. Based on the nested reduction, we design several quick algorithms to find a different reduct when one reduct is already given. Here 'different' means the reducts obtained on different parameters. All these algorithms are helpful to quickly finding a proper reduct in parameterized rough set models. Finally, the numerical experiments demonstrate the feasibility and the affectivity of the nested reduction approach.