L-fuzzy concept analysis using fuzzy categories

The purpose of this paper is to investigate topics in fuzzy concept analysis using the theory of arrow and fuzzy categories. Our approach deals with the original, fuzzy data using fuzzy subsets in order to obtain a notion of fuzzy concepts, fuzzy concept lattices, and attribute implications between fuzzy sets of attributes. Our approach is general in the sense that we only require a Heyting algebra together with a t-norm like operation as the lattice of truth values instead of the unit interval [0,1] or a substructure thereof. As a consequence, regular concept analysis, i.e., concept analysis based on regular sets and the Boolean truth values is just a special case of our approach. In addition, our approach differs from a lot of approaches in the literature, which usually first apply some kind of defuzzyfication, e.g., α-cuts, and then apply concept analysis in the classical (crisp) sense.