Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Xie, Mingchun
Institution: Sichuan University
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213405943