Completeness of the Fuzzy Order on Fuzzy Numbers

Mingchun Xie

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arXiv · 2609.16052

Completeness of the Fuzzy Order on Fuzzy Numbers

Mingchun Xie

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Abstract

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.

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BibTeX RIS

Mingchun Xie. 2026-09-13. Completeness of the Fuzzy Order on Fuzzy Numbers. https://arxiv.org/abs/2609.16052

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