On matchable subsets in abelian groups and their linear analogues

TL;DR

This paper explores matchable subsets in abelian groups and their linear analogues, introducing new concepts like matching matrices and studying conditions for acyclic matchings and local matchability, combining combinatorics and linear algebra.

Contribution

It introduces the notions of matching matrices and linear local matching properties, providing new criteria and bounds for matchings in abelian groups and vector spaces.

Findings

Necessary conditions for acyclic matchings in abelian groups.

Dimension criteria for linear locally matchable bases.

Upper bounds for primitive subspace dimensions.

Abstract

In this paper, we introduce the notions of matching matrices in groups and vector spaces, which lead to some necessary conditions for existence of acyclic matching in abelian groups and its linear analogue. We also study the linear local matching property in field extensions to find a dimension criteria for linear locally matchable bases. Moreover, we define the weakly locally matchable subspaces and we investigate their relations with matchable subspaces. We provide an upper bound for the dimension of primitive subspaces in a separable field extension. We employ MATLAB coding to investigate the existence of acyclic matchings in finite cyclic groups. Finally, a possible research problem on matchings in n-groups is presented. Our tools in this paper mix combinatorics and linear algebra.