Triadic fuzzy Galois connections as ordinary connections

Abstract-The paper presents results on representation of the basic structures related to ternary fuzzy relations by the structures related to ordinary ternary relations, such as Galois connections, closure operators, and trilattices (structures of maximal Cartesian subrelations). These structures appear as the fundamental structures in relational data analysis such as formal concept analysis or association rules. We prove several representation theorems that allow us to automatically transfer some of the known results from the ordinary case to fuzzy case. The transfer is demonstrated by examples. I. INTRODUCTION Relations play a fundamental role in mathematics, computer science, and their applications. Many results about ordinary relations have been generalized to the setting of fuzzy relations in the past. There has always been a fundamental question of how the various fuzzifications are related to the ordinary notions and results. Needless to say, this question is important both from a practical and theoretical point of view and is treated to some extent in textbooks, see e.g. In this paper we deal with basic structures associated to ternary relations that appear as fundamental ones in the methods of relational data analysis, namely the closure-like structures such as Galois connections, closure operators, structures of their fixpoints and the like. Such structures appear e.g. in formal concept analysis The most common way of looking at the relationship between ordinary notions and their fuzzy counterparts is in terms of a-cuts of fuzzy relations (see e.g. [15]) but there are additional possible views at the question as well. One of them, utilized in this paper, is provided in [3, Section 3.1.2]. Our paper is organized as follows. We first provide preliminaries in Section II. In Section III, we introduce the Galoi