Answers to some questions about Zadeh's extension on metric spaces

TL;DR

This paper demonstrates that Zadeh's extension of certain contractions in metric spaces does not necessarily preserve the contraction property under the Skorokhod metric, providing a negative answer to previously posed open problems.

Contribution

It provides a counterexample showing that Zadeh's extension can fail to be a contraction under the Skorokhod metric, addressing open questions in fuzzy set theory.

Findings

Existence of a contraction whose Zadeh's extension is not a contraction under the Skorokhod metric

Negative answer to Problems 5.8 and 5.12 from prior work

Clarification of limitations in Zadeh's extension in metric spaces

Abstract

This paper shows that there exists a contraction whose Zadeh's extension is not a contraction under the Skorokhod metric, answering negatively Problems 5.8 and 5.12 posted in [Jard\'{o}n, S\'{a}nchez, and Sanchis, Some questions about Zadeh's extension on metric spaces, Fuzzy Sets and Systems, 2018].