The paint pot problem and common multiples in monoids

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Applicable Algebra in Engineering Communication and Computing Pub Date : 2023-05-13 DOI:10.1007/s00200-023-00606-6

Hans Zantema, Vincent van Oostrom

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Abstract

Illustrated by a problem on paint pots that is easy to understand but hard to solve, we investigate whether particular monoids have the property of common right multiples. As one result we characterize generalized braid monoids represented by undirected graphs, being a subclass of Artin–Tits monoids. Stated in other words, we investigate to which graphs the old Garside result stating that braid monoids have the property of common right multiples, generalizes. This characterization also follows from old results on Coxeter groups and the connection between finiteness of Coxeter groups and common right multiples in Artin–Tits monoids. However, our independent presentation is self-contained up to some basic knowledge of rewriting, and also applies to monoids beyond the Artin–Tits format. The main new contribution is a technique to prove that the property of common right multiples does not hold, by finding a particular model, in our examples all being finite.

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油漆罐问题和一元群的公倍数

以一个容易理解但难于解决的油漆罐问题为例，研究了特定模群是否具有公右倍数的性质。作为一个结果，我们刻画了由无向图表示的广义辫形monoids，它是Artin-Tits monoids的一个子类。换句话说，我们研究了关于辫形模群具有公右倍数性质的旧Garside结果推广到哪些图上。这一性质也来自于关于Coxeter群的旧结果以及Artin-Tits一元群中Coxeter群的有限性与公右倍数之间的联系。然而，我们的独立表示是自包含的，直到一些重写的基本知识，也适用于超越art - tits格式的monoids。主要的新贡献是一种技术来证明公右乘的性质不成立，通过找到一个特定的模型，在我们的例子都是有限的。

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来源期刊

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.