Gradation of Fuzzy Preconcept Lattices

Alexander Šostak,Ingrīda Uļjane,Māris Krastiņš

doi:10.3390/axioms10010041

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.

Highlights

Formal concept analysis, or just concept analysis for short, was developed mainly in eighties of the previous century by R
In the two theorems, we prove the basic properties of D-graded fuzzy preconcept lattices
Noticing the limitation of the concept lattices in case of a fuzzy context in view of the possible applications, especially for “real world” problems, we introduce here a very general notion of a preconcept, and on the other hand resrtrict it by assigning to a preconcept a certain “degree of its conceptuality”

Summary

Introduction

Just concept analysis for short, was developed mainly in eighties of the previous century by R. The set of all such pairs in a given context (X, Y, R) endowed with a certain partial order makes a lattice, called a concept lattice, the principal object of research in concept analysis. We define fuzzy preconcepts, introduce partial order relation on the family of all fuzzy preconcepts of a given fuzzy context (X, Y, L, R), and show that the resulting structure (P(X, Y, L, R), ) is a lattice. In case of crisp object and attribute sets A, B and under assumption that either (†∗) or (†R) holds, the degree of the conceptuality of the pair (A, B) is D(A, B) = x∈A,y∈BR(x, y). X∈Xa,y∈Y1 (a → R(x, y)) ∧ x∈Xa a → (b → y∈Yb R(x, y))

D-Graded Preconcept Lattices

Conceptional Hull and Conceptional Kernel of a Fuzzy Preconcept

Findings

Conclusions