Characterization and reduction of concept lattices through matroid theory

Concept lattice theory is an efficient tool for data analysis. Reduction of attributes and objects for a context is an important issue in concept lattice theory. We explore the relationships between matroidal spaces and concepts for a given context. With the assistance of matroid theory, we conduct the attribute reduction and object reduction for a context and formulate the concepts for a given context. As a generalization of the linear independence in vector spaces, matroids provide well established platforms for some algorithms such as greedy algorithm.In this paper, we mainly propose a notion of matroidal space by the family of circuits of a matroid. These relationships characterize the constructions of not only concepts, but also concept lattices. Meanwhile, we demonstrate that reduction of attributes and objects proposed in this paper preserve the original hierarchy order in the concept lattice for a given context.