Rough matroids based on coverings
This work addresses foundational issues in rough set theory, specifically for researchers in mathematical logic and combinatorial optimization, but it is incremental as it builds on prior work by Zhu and Wang.
The paper tackles the problem of extending rough matroids from relations to coverings, a generalization in rough set theory, by investigating properties of definable sets and proposing a new formulation, with results including the lattice structure of definable sets and an equivalent formulation for rough matroids based on coverings.
The introduction of covering-based rough sets has made a substantial contribution to the classical rough sets. However, many vital problems in rough sets, including attribution reduction, are NP-hard and therefore the algorithms for solving them are usually greedy. Matroid, as a generalization of linear independence in vector spaces, it has a variety of applications in many fields such as algorithm design and combinatorial optimization. An excellent introduction to the topic of rough matroids is due to Zhu and Wang. On the basis of their work, we study the rough matroids based on coverings in this paper. First, we investigate some properties of the definable sets with respect to a covering. Specifically, it is interesting that the set of all definable sets with respect to a covering, equipped with the binary relation of inclusion $\subseteq$, constructs a lattice. Second, we propose the rough matroids based on coverings, which are a generalization of the rough matroids based on relations. Finally, some properties of rough matroids based on coverings are explored. Moreover, an equivalent formulation of rough matroids based on coverings is presented. These interesting and important results exhibit many potential connections between rough sets and matroids.