Q的加法子群：对数描述与交集构型

中文摘要 AI 辅助

我们发展了一个对数框架来研究有理数加法群的子群格。通过用素数指数序列编码正有理数，可以得到$\mathbb{Q}^+$与以素数为索引的有限支撑整值序列之间的双射。这一对应关系通过对数最大公约数扩展到$(\mathbb{Q},+)$的任意子群，从而根据$\mathbb{Z}\cup \{-\infty\}$中的最终非正序列对子群进行分类。在该框架内，子群的包含、和、积与交集具有简单的坐标式描述，为子群格提供了透明的解释，并恢复了若干经典结果，如同构意义下的子群分类。利用这一视角，我们获得了$(\mathbb{Q},+)$的子群中可实现交集构型的完整刻画。对于秩构型，可实现性由递减条件以及可能秩和最小零集的限制来刻画，并且在有限支撑情形下，还需满足额外的基数条件。对于二元构型，这归结为：在无限支撑情形下，每个递减构型都是可实现的；而在有限支撑情形下，则需要上述基数条件。

英文摘要

We develop a logarithmic framework to study the lattice of subgroups of the additive group of rational numbers. By encoding positive rationals via their prime exponent sequences, one obtains a bijection between $\mathbb{Q}^+$ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of $(\mathbb{Q},+)$ through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in $\mathbb{Z}\cup \{- \infty\}$. Within this framework, subgroup inclusion, sum, product, and intersection admit simple coordinatewise descriptions, providing a transparent interpretation of the subgroup lattice and recovering several classical results such as the subgroup classification up to isomorphism. Exploiting this perspective, we obtain a complete characterization of the intersection configurations realizable in subgroups of $(\mathbb{Q},+)$. For rank configurations, realizability is characterized by the decreasing condition together with restrictions on the possible ranks and on the minimal zero sets, and, in the finite-support case, by an additional cardinality condition. For binary configurations, this reduces to saying that every decreasing configuration is realizable in the infinite-support case, whereas in the finite-support case the mentioned cardinality condition is required.

发表机构

• Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）

机构由 AI 辅助整理，请以论文原文为准。

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27 pages, 4 figures