Non-unique factorizations, land surveying and electricity

TL;DR

This paper bridges non-unique factorizations with graph theory by translating algebraic properties into Cayley graph characteristics, revealing new insights into the structure of abelian groups and their subsets.

Contribution

It introduces a novel connection between factorization properties and Cayley graph structures, specifically characterizing half factorial and weak half factorial subsets via geodetical and voltage digraphs.

Findings

Half factorial subsets correspond to geodetical Cayley digraphs.

Weakly half factorial subsets satisfy Kirchhoff's Voltage Law in associated voltage digraphs.

New framework links algebraic factorization properties with graph theoretical concepts.

Abstract

Non-unique factorizations theory, which started in algebraic number theory, over the years has expanded into several areas of mathematics. Here, we propose yet another branching. We show that some concepts of factorizations theory, such as half factorial and weak half factorial properties can be translated via Cayley graphs into graph theory. It is proved, that subset S of abelian group G is half factorial, if and only if the Cayley digraph Cay(G; S) is geodetical, e.g., simple paths connecting a fixed pair of vertices have the same length. Further, it is shown that the voltage digraph naturally arising from subset S of group G satisfies Kirchoff's Voltage Law exactly when S is weakly half factorial. In the concluding remarks, some loosely formulated ideas for further research are presented..