有限群的幂半环的有限基问题
The finite basis problem for the power semirings of finite groups
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中文总结 AI 辅助
该研究明确有限群$G$的幂半环$\boldsymbol{\textit{P}}(G)$无恒等式有限基当且仅当群的阶不小于3，完成了有限群幂半环关于有限基性质的分类。
中文摘要 AI 辅助
对于任意群$G$，$G$的所有非空子集构成的集合在集合论的并运算与元素乘法下形成一个加法幂等半环，称为$G$的幂半环，记为$\boldsymbol{\textit{P}}(G)$。我们证明：有限群$G$的幂半环$\boldsymbol{\textit{P}}(G)$没有恒等式的有限基当且仅当$|G| \boldsymbol{\text{≥}} 3$。这完成了有限群的幂半环关于有限基性质的分类。
英文摘要
For any group $G$, the set of all nonempty subsets of $G$ forms an additively idempotent semiring under set-theoretic union and elementwise multiplication, called the power semiring of $G$ and denoted by $\mathcal{P}(G)$. We prove that for a finite group $G$, $\mathcal{P}(G)$ has no finite basis for its identities if and only if $|G| \geq 3$. This completes the classification of the power semirings of finite groups with respect to the finite basis property.
