Galois connections, concept lattices, and related structures in fuzzy logic
by
Radim Belohlavek
Dept. Computer Science, Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic

The main purpose of the talk is to present a theory of Galois connections, concept lattices, and related structures from the point of view of fuzzy logic. Fuzzy logic and fuzzy set theory originated by Zadeh in 1965 has been substantially developed recently. Contrary to classical logic that works with two truth values (0 and 1), fuzzy logic works with an ordered scale of truth values (truth degrees). This makes it possible to grasp the intuition that propositions of natural language are not always fully true or fully false; rather, they are true to a certain degree (examples: "100 is a big number", "it is very cold"). Formalization of this idea leads to logical calculi for so-called approximate reasoning and to the theory of fuzzy sets (classical logic and set theory are special cases). "Fuzzy approach" has been applied to various disciplines and theories; there are numerous commercial applications of fuzzy logic. We show a way to generalize the theory of Galois connections, concept lattices, and related structures to fuzzy setting. The generalization is well-motivated: from the point of view of the theory of concept lattices, the present generalization makes it possible to handle vagueness in data (object-attribute relations, formal concepts etc.). The issues that will be discussed are: the notion of a fuzzy Galois connection; fuzzy Galois connections induced by binary fuzzy relations; relationship between fuzzy Galois connections and (non-fuzzy) Galois connections; fixed points of fuzzy Galois connections and the notion of a fuzzy concept lattice; fuzzy closure operators and closure systems; fuzzy order, the structure of fixed points of fuzzy Galois connections, and the main theorem of fuzzy concept lattices; some issues that are degenerated in classical case (logical precision, similarity). Briefly discussed will be the connection of the present approach to other approaches (fuzzy concept lattices: A.Burusco+R.Fuentes-Gonzales, S.Pollandt; fuzzy closure operators: Italian and Spanish groups) and some directions of future research.

Date received: January 23, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagi-10.