发表机构
Stellenbosch University; University of Minho（斯泰伦博斯大学; 米尼奥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文系统刻画了素数幂模幺半群间的乘法同态，分三步确定单位群同态、扩展条件及完整同态集，并建立与低层级单位群构造的幺半群的显式同构。
AI 中文摘要
我们对素数幂模幺半群 $(\nmathbb{Z}/p^e\mathbb{Z},\cdot)$ 与 $(\nmathbb{Z}/q^f\mathbb{Z},\cdot)$ 之间的乘法幺半群同态进行了系统性研究，其中 $p,q$ 为任意素数，$e,f\geq 1$ 为整数。我们的方法分三个阶段进行。首先，我们确定相应单位群之间的群同态同态集 $\mathbf{Hom}(U_{p^e},U_{q^f})$，并展示描述其群结构的显式同构。然后，我们识别单位群同态中能扩展为 $(\nmathbb{Z}/p^e\mathbb{Z},\cdot)$ 上完整幺半群同态的子集 $\mathcal{H}_{p^e,q^f,v}$，通过显式整除条件刻画它并描述其代数结构。最后，我们给出 $\mathbf{Hom}_{\mathbf{Mon}}((\mathbb{Z}/p^e\mathbb{Z},\cdot), (\mathbb{Z}/q^f\mathbb{Z},\cdot))$ 的完整描述，并建立该同态集与由低层级模环单位群自然构造的幺半群之间的显式幺半群同构。
英文摘要
We undertake a systematic study of multiplicative monoid morphisms between prime-power modular monoids $(\mathbb{Z}/p^e\mathbb{Z},\cdot)$ and $(\mathbb{Z}/q^f\mathbb{Z},\cdot)$, for arbitrary primes $p,q$ and integers $e,f\geq 1$. Our approach proceeds in three stages. We first determine the hom-set $\mathbf{Hom}(U_{p^e},U_{q^f})$ of group morphisms between the corresponding unit groups and exhibit explicit isomorphisms describing its group structure. We then identify the subset $\mathcal{H}_{p^e,q^f,v}$ of unit group morphisms that extend to full monoid morphisms on $(\mathbb{Z}/p^e\mathbb{Z},\cdot)$, characterizing it via explicit divisibility conditions and describing its algebraic structure. Finally, we give a complete description of $\mathbf{Hom}_{\mathbf{Mon}}((\mathbb{Z}/p^e\mathbb{Z},\cdot), (\mathbb{Z}/q^f\mathbb{Z},\cdot))$ and establish an explicit monoid isomorphism between this hom-set and a naturally defined monoid built from unit groups of lower-level modular rings.
Comments19
