PreprintCC BY 4.0arXiv
Beyond left-continuity: t-norms on the fuzzy unit interval induced by pairs of t-norms
Abstract
We define a binary operation T1⋆ T2 on the fuzzy unit interval I by extending the natural action of T1 and T2 on step functions. When both T1 and T2 are left-continuous, this operation coincides with the tensor product of the corresponding quantales and therefore yields a t-norm on I. Without left-continuity, the situation becomes more subtle. We prove that if at least one of the two t-norms is continuous, then T1⋆ T2 is still a t-norm on I. We also show by counterexample that continuity cannot simply be dropped. The operation DN fails to be associative. On the other hand, TD⋆ TD is a t-norm despite TD being neither continuous nor left-continuous, indicating that continuity is not a necessary condition.
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BibTeX
@misc{wu2026beyond,
title = {{Beyond left-continuity: t-norms on the fuzzy unit interval induced by pairs of t-norms}},
author = {Rongjun Wu and Jie Sun and Hongliang Lai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40678},
eprint = {2610.40678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40678},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond left-continuity: t-norms on the fuzzy unit interval induced by pairs of t-norms
AU - Rongjun Wu
AU - Jie Sun
AU - Hongliang Lai
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.40678
UR - https://arxiv.org/abs/2610.40678
AB - We define a binary operation T_1⋆ T_2 on the fuzzy unit interval I by extending the natural action of T_1 and T_2 on step functions. When both T_1 and T_2 are left-continuous, this operation coincides with the tensor product of the corresponding quantales and therefore yields a t-norm on I. Without left-continuity, the situation becomes more subtle. We prove that if at least one of the two t-norms is continuous, then T_1⋆ T_2 is still a t-norm on I. We also show by counterexample that continuity cannot simply be dropped. The operation DN fails to be associative. On the other hand, T_D⋆ T_D is a t-norm despite T_D being neither continuous nor left-continuous, indicating that continuity is not a necessary condition.
ER -