Completeness of the Fuzzy Order on Fuzzy Numbers

This paper investigates the completeness properties of the fuzzy order on the fuzzy number space ${\mathbb{E}}^1$ within the framework of $[0,1]$-enriched categories. $S$-convergence under the three fundamental continuous t-norms is characterized, and inclusion relations among the induced open ball topologies are established. It is shown that $({\mathbb{E}}^1,P)$ is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas $({\mathbb{E}}^1,P)$ fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization $({\mathbb{E}}^1,S)$ is Smyth complete if and only if the underlying continuous t-norm is not isomorphic to the Łukasiewicz t-norm.