On an Open Problem in Bottleneck Algebra

TAN Yi-jia

doi:10.13371/j.cnki.chin.q.j.m.2025.03.008

Chinese Quarterly Journal of Mathematics >

2025 , Vol. 40 >Issue 3: 324 - 330

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2025.03.008

On an Open Problem in Bottleneck Algebra

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School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China

TAN Yi-jia (1962-), male, native of Xianning, Hubei, professor of Fuzhou University, engages in algebra, fuzzy mathematics, etc.

Received date: 2025-02-28

Online published: 2025-09-30

Supported by

Supported by National Natural Science Foundation of China (Grant Nos. 11771004 and 11971111).

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Abstract

A bottleneck algebra is a linearly ordered set (B,≤) with two operations a⊕b=max{a,b} and a⊗b=min{a,b}. A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way. In 1996, Cechl´arov´a and Pl´avka posed an open problem: Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2B-independent. In this paper, we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2B-independent and answer this open problem.

Key words： Bottleneck algebra; Vector; 2B-independence

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TAN Yi-jia . On an Open Problem in Bottleneck Algebra[J]. Chinese Quarterly Journal of Mathematics, 2025 , 40(3) : 324 -330 . DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.008

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