Fuzzy Galois connections on fuzzy sets
Javier Gutiérrez García, Hongliang Lai, Lili Shen
TL;DR
The paper develops an accessible, order-theoretic treatment of quantale-valued preorders on fuzzy sets, grounded in the quantaloid $\mathcal{D}\mathfrak{Q}$ and enriched-category ideas. It extends fuzzy Galois theory to $\mathfrak{Q}$-preordered $\mathfrak{Q}$-subsets by introducing $\mathfrak{Q}$-distributors, $\mathfrak{Q}$-Galois connections, $\mathfrak{Q}$-polarities, and (dual) $\mathfrak{Q}$-axialities, and shows deep equivalences among these notions. Completeness, tensors/cotensors, and powerset constructions play central roles, with fixed-point results like MacNeille completions arising from polarity structures. The framework unifies fuzzy-context analysis (FCA/RST) with enriched-categorical concepts, providing a robust toolkit for reasoning about fuzzy relations, preorders, and adjunctions in a purely order-theoretic language. This yields a versatile bridge between fuzzy set theory and quantaloid-enriched category theory, with potential applications in formal concept analysis and rough set theory on fuzzy data.
Abstract
In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the readership of fuzzy set theorists, with particular attention paid to fuzzy Galois connections between preordered fuzzy sets.