Yoneda completeness and flat completeness of ordered fuzzy sets

TL;DR

This paper explores the concepts of Yoneda and flat completeness in ordered fuzzy sets valued in a quantale derived from the unit interval with a continuous triangular norm, extending classical order theory to fuzzy contexts.

Contribution

It establishes the relationship between Yoneda and flat completeness in fuzzy ordered sets and identifies conditions under which the converse implication holds.

Findings

Flat completeness implies Yoneda completeness.

The converse holds only for Lukasiewicz or product t-norms.

Provides conditions for completeness in fuzzy ordered structures.

Abstract

This paper studies Yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm. Both of these notions are natural extension of directed completeness in order theory to the fuzzy setting. Yoneda completeness requires every forward Cauchy net converges (has a Yoneda limit), while flat completeness requires every flat weight (a counterpart of ideals in partially ordered sets) has a supremum. It is proved that flat completeness implies Yoneda completeness, but, the converse implication holds only in the case that the related triangular norm is either isomorphic to the Lukasiewicz t-norm or to the product t-norm.