关于Bottleneck代数的一个公开问题

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

数学季刊 ›› 2025, Vol. 40 ›› Issue (3): 324-330.doi: 10.13371/j.cnki.chin.q.j.m.2025.03.008

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关于Bottleneck代数的一个公开问题

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

收稿日期:2025-02-28 出版日期:2025-09-30 发布日期:2025-09-30
作者简介:TAN Yi-jia (1962-), male, native of Xianning, Hubei, professor of Fuzhou University, engages in algebra, fuzzy mathematics, etc.
基金资助:
Supported by National Natural Science Foundation of China (Grant Nos. 11771004 and 11971111).

On an Open Problem in Bottleneck Algebra

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

Received:2025-02-28 Online:2025-09-30 Published:2025-09-30
About author:TAN Yi-jia (1962-), male, native of Xianning, Hubei, professor of Fuzhou University, engages in algebra, fuzzy mathematics, etc.
Supported by:
Supported by National Natural Science Foundation of China (Grant Nos. 11771004 and 11971111).

摘要/Abstract

摘要： 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.

关键词:

Bottleneck algebra, Vector, 2B-independence

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

中图分类号:

O153.1

谭宜家. 关于Bottleneck代数的一个公开问题[J]. 数学季刊, 2025, 40(3): 324-330.

TAN Yi-jia. On an Open Problem in Bottleneck Algebra[J]. Chinese Quarterly Journal of Mathematics, 2025, 40(3): 324-330.

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关于Bottleneck代数的一个公开问题

On an Open Problem in Bottleneck Algebra

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