PreprintCC BY-NC-ND 4.0arXiv
Associativity of discretized triangular norms
Abstract
In this article, we mainly investigate when the binary operation T[n]:[0, 1]2→ [0, 1] satisfies the associative law and thus becomes a triangular norm where T[n](x,y)={ll fn(T(fn(x),fn(y))), &if (x,y)∈[0,1)2, min{x,y}, &otherwise, . fn:[0,1]→{0, 1/n, 2/n, …, (n−1)/n, 1} given by fn(x)=1/n⌊ nx ⌋ with n a positive integer and T:[0, 1]2→ [0, 1] is a triangular norm. We present some sufficient conditions for T[n] to be triangular norms, and for T[n] to be never associative when T is strictly monotonic.
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BibTeX
@misc{zhang2026associativity,
title = {{Associativity of discretized triangular norms}},
author = {Di Zhang and Xue-ping Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14072},
eprint = {2610.14072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14072},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Associativity of discretized triangular norms
AU - Di Zhang
AU - Xue-ping Wang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.14072
UR - https://arxiv.org/abs/2610.14072
AB - In this article, we mainly investigate when the binary operation T^[n]:[0, 1]^2→ [0, 1] satisfies the associative law and thus becomes a triangular norm where T^[n](x,y)={ll f_n(T(f_n(x),f_n(y))), &if (x,y)∈[0,1)^2, min{x,y}, &otherwise, . f_n:[0,1]→{0, 1/n, 2/n, …, (n−1)/n, 1} given by f_n(x)=1/n⌊ nx ⌋ with n a positive integer and T:[0, 1]^2→ [0, 1] is a triangular norm. We present some sufficient conditions for T^[n] to be triangular norms, and for T^[n] to be never associative when T is strictly monotonic.
ER -