Gradation of Fuzzy Preconcept Lattices

Abstract

Noticing certain limitations of concept lattices in the fuzzy context, especially in view of their practical applications, in this paper, we propose a more general approach based on what we call graded fuzzy preconcept lattices. We believe that this approach is more adequate for dealing with fuzzy information then the one based on fuzzy concept lattices. We consider two possible gradation methods of fuzzy preconcept lattice—an inner one, called D-gradation and an outer one, called M-gradation, study their properties, and illustrate by a series of examples, in particular, of practical nature.

Fuzzy Algebras of Concepts

“…These contributions extend previous results on the gradation of the powerset lattice [7], providing a more comprehensive study of the intermediate algebraic structures that arise between the preconcepts and the formal concepts. In this sense, our work may help in the understanding of the underlying relationship between the crisp notions of pre-, proto-, and semiconcepts, as well as in providing a panoramic view of the conceptual space in the fuzzy paradigm.…”

“…In the Introduction, it has been mentioned that some previous work [7] has dealt with the issue of concept grading, although without taking into account intermediate structures of preconcepts, protoconcepts, or semiconcepts. Although preconcepts are mentioned in that work, in reality this terminology refers to the powerset L G × L M .…”

“…Recently, Šostak et al [7] have presented an approach to the fuzzy grading of preconcepts, based on quantales (complete lattices equipped with an associative product operation, distributive with respect to arbitrary joins). The preconcept terminology in that work, does not coincide with the traditional one in FCA [8], although it is consistent with their proposal: for the authors, the set of preconcepts coincides exactly with the powerset of G × M, and the preprefix only indicates that these elements are used as generalisations of formal concepts.…”