AI 中文总结
本文研究 $0$-单位 $3$-度循环有向图的自同态幺半群的自同构群，通过建立无 $2$-圈单位循环有向图存在诱导有向圈的自同态的充要条件，显式确定了正规化子群 $U_S(\mathbb{Z}_n)$，解决了该族的未决问题。
AI 中文摘要
设 $G=\Cay(\mathbb{Z}_n,S)$ 为连接集 $S=\{0,s,t\}$ 的 $3$-度循环有向图，其中 $s,t\in\mathbb{Z}_n^*$ 且 $t\neq\pm s$。确定其自同态幺半群的自同构群归结为确定正规化 $\End(G)$ 的子群 $U_S(\mathbb{Z}_n)\leq\mathbb{Z}_n^*$。本文首先建立了无 $2$-圈的单位循环有向图存在自同态且其像诱导有向圈的充要条件。利用该刻画，我们显式确定了先前未解决的 $0$-单位 $3$-度族中的 $U_S(\mathbb{Z}_n)$。
英文摘要
Let $G=\Cay(\mathbb{Z}_n,S)$ be a $3$-valent circulant digraph with connection set $S=\{0,s,t\}$, where $s,t\in\mathbb{Z}_n^*$ and $t\neq\pm s$. Determining the automorphism group of its endomorphism monoid reduces to determining the subgroup $U_S(\mathbb{Z}_n)\leq\mathbb{Z}_n^*$ that normalizes $\End(G)$. In this paper, we first establish necessary and sufficient conditions for a unit circulant digraph without $2$-cycles to admit an endomorphism whose image induces a directed cycle. Using this characterization, we explicitly determine $U_S(\mathbb{Z}_n)$ for the previously unresolved $0$-unit $3$-valent family.
