Abstract

A decomposition of an integer n as a sum of several fractions is called faithful if no positive integer less than n can be obtained by replacing some of the numerators with smaller nonnegative integers. In this note, we prove the existence of faithful decompositions of all positive integers n, and determine the minimum number of fractions of the faithful decomposition. Here we also illustrate how the results we establish can be naturally translated into the language of Puiseux monoids.

中文翻译：

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关于整数的分数分解的一个注记

摘要

如果无法通过用较小的非负整数替换某些分子而无法获得小于n的正整数，则将整数n分解为几个分数的总和称为分解。在此注释中，我们证明了所有正整数n的忠实分解的存在，并确定了忠实分解的分数的最小数目。在这里，我们还说明了如何将建立的结果自然地翻译为Puiseux monoids语言。