非唯一因子分解、土地测量和电力
Non-unique factorizations, land surveying and electricity
摘要：
非唯一因子分解理论起源于代数数论，多年来已经扩展到数学的几个领域。在这里，我们提出另一个分支。我们证明了因子分解理论中的一些概念，如半因子和弱半因子性质，可以通过Cayley图转化为图论。证明了阿贝尔群G的子集S是半阶乘的当且仅当Cayley有向图Cay（G; S）是测地的，例如，连接固定顶点对的简单路径具有相同的长度。进一步证明了当S是弱半阶乘时，由群G的子集S自然产生的电压有向图完全满足基尔霍夫电压定律。在结束语中，提出了一些松散的想法，为进一步的研究。
英文摘要：
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..
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
