Necessity of the non-Archimedean condition

Determine whether the non-Archimedean condition is necessary for a George–Veeramani fuzzy metric space to satisfy the implication that every strictly fuzzy ψ-contractive sequence is Cauchy, specifically by determining whether a non-strong George–Veeramani fuzzy metric space can exist in which every strictly fuzzy ψ-contractive sequence is nevertheless Cauchy.

Background

The paper explains that non-Archimedean structure permits the control of adjacent sequence terms to propagate to arbitrary pairs of terms, which is the mechanism used in known positive Cauchy results for fuzzy ψ-contractive sequences.

The constructed counterexample is not strong and demonstrates that strict fuzzy ψ-contractivity does not imply Cauchyness in at least one non-Archimedean-free setting. The authors do not determine whether non-Archimedeanness is genuinely necessary whenever the implication does hold, or whether some non-strong George–Veeramani fuzzy metric space might still force every strictly fuzzy ψ-contractive sequence to be Cauchy.

References

We conclude by proposing two open problems arising naturally from our investigation. Problem 3. Does there exist a strictly fuzzy $\psi$-contractive sequence that is not Cauchy in a GV-fuzzy metric space equipped with the product t-norm or the {\L}ukasiewicz t-norm? In other words, does the choice of the continuous t-norm affect the validity of the statement that strictly fuzzy $\psi$-contractive implies Cauchy? Problem 4. Is the non-Archimedean condition a necessary structural requirement for the statement that strictly fuzzy $\psi$-contractive implies Cauchy to hold in a GV-fuzzy metric space? That is, does there exist a non-strong GV-fuzzy metric space in which every strictly fuzzy $\psi$-contractive sequence is nevertheless Cauchy?

— On fuzzy contractions in fuzzy metric spaces  (2609.03372 - Xu et al., 3 Sep 2026) in Section 4, Concluding remarks, Problem 4