AI 中文总结
该研究证明映射度集在特定运算下保持稳定，实现了由ℤ的加法子半群经有限次和与积得到的集合作为映射度集，拓展了实现问题研究并关联相关学者提出的问题。
AI 中文摘要
我们证明，映射度集在与包含0的ℤ的有限子集相乘、以及与ℤ的加法子半群经有限次和与积得到的集合相乘时保持稳定。特别地，后一类集合中的每一个都可作为映射度集。由此，我们得到一大类无限映射度集，包括从0开始的算术级数的有限并。这些结果拓展了此前关于实现问题的研究，且与Neofytidis、Wang和Wang提出的问题相关。
英文摘要
We prove that mapping degree sets are stable under multiplication by finite subsets of $\mathbb Z$ containing $0$ and by sets obtained from additive submonoids of $\mathbb Z$ through finitely many sums and products. In particular, every set of the latter type occurs as a mapping degree set. As a consequence, we obtain a broad family of infinite mapping degree sets, including finite unions of arithmetic progressions starting at $0$. These results extend previous work on the realization problem and are related to a question posed by Neofytidis, Wang, and Wang.
Comments15 pages, no figures
