Toward reduction of formal fuzzy context

We consider the problem of characterizing and computing a minimal context for a given fuzzy context. That is, for a given binary fuzzy relation between a set of objects and a set of attributes, we want to obtain a new fuzzy relation in such a way that its concept lattice is isomorphic to the concept lattice of the original fuzzy relation and such that the new fuzzy relation is minimal. It turns out that the essence of this problem may be rephrased as the problem of finding bases of fuzzy closure systems. In the Boolean case, the problem is relatively simple and its solution is well known. In a fuzzy setting, the problem is more complex, basically because there are two generating operations involved: intersection, which is the only operation involved in the Boolean case, and shift, which is degenerate in the Boolean case. In this paper, we present some first results, examples, and issues for future research in this problem. Proofs and some technical material is left to a full version of this paper.