Matchings and product growth in modular abelian independence groups
Abstract
We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group G, we develop an analogue of the e-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a finite-rank submonoid H satisfying 1<ρ(H)<ρ(G), where ρ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy--Davenport theorem: we define a parameter μ(G) and prove that ρ(XY)≥ min{μ(G),ρ(X)+ρ(Y)−1} for all nonempty finite-rank subsets X,Y of G. Furthermore, ρ(XY) is shown to be controlled from below by a submonoid of G that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.
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BibTeX
@misc{aliabadi2026matchings,
title = {{Matchings and product growth in modular abelian independence groups}},
author = {Mohsen Aliabadi and Jozsef Losonczy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18340},
eprint = {2608.18340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18340},
}
RIS
TY - UNPB
TI - Matchings and product growth in modular abelian independence groups
AU - Mohsen Aliabadi
AU - Jozsef Losonczy
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2608.18340
UR - https://arxiv.org/abs/2608.18340
AB - We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group G, we develop an analogue of the e-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a finite-rank submonoid H satisfying 1<ρ(H)<ρ(G), where ρ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy--Davenport theorem: we define a parameter μ(G) and prove that ρ(XY)≥ min{μ(G),ρ(X)+ρ(Y)−1} for all nonempty finite-rank subsets X,Y of G. Furthermore, ρ(XY) is shown to be controlled from below by a submonoid of G that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.
ER -