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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 𝔼1 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 (𝔼1,P) is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas (𝔼1,P) fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization (𝔼1,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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Cite

BibTeXRIS
BibTeX
@misc{xie2026completeness,
  title = {{Completeness of the Fuzzy Order on Fuzzy Numbers}},
  author = {Mingchun Xie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16052},
  eprint = {2609.16052},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16052},
}
RIS
TY  - UNPB
TI  - Completeness of the Fuzzy Order on Fuzzy Numbers
AU  - Mingchun Xie
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.16052
UR  - https://arxiv.org/abs/2609.16052
AB  - This paper investigates the completeness properties of the fuzzy order on the fuzzy number space 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 (𝔼^1,P) is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas (𝔼^1,P) fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization (𝔼^1,S) is Smyth complete if and only if the underlying continuous t-norm is not isomorphic to the Łukasiewicz t-norm.
ER  -