Additive subgroups of Q: logarithmic description and intersection configurations
Abstract
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 ℚ+ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of (ℚ,+) through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in ℤ∪ {− ∞}. 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 (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.
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BibTeX
@misc{delgado2026additive,
title = {{Additive subgroups of Q: logarithmic description and intersection configurations}},
author = {Jordi Delgado and Antoni Massegú and Enric Ventura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37357},
eprint = {2609.37357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37357},
}
RIS
TY - UNPB
TI - Additive subgroups of Q: logarithmic description and intersection configurations
AU - Jordi Delgado
AU - Antoni Massegú
AU - Enric Ventura
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37357
UR - https://arxiv.org/abs/2609.37357
AB - 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 ℚ^+ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of (ℚ,+) through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in ℤ∪ {− ∞}. 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 (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.
ER -