AI 中文总结
本文研究有限群幂半环簇的格区间，证明阶至少三的有限群的非空幂半环之上、有限生成簇之下的所有簇均非有限基，并给出完全幂半环有限基的充要条件。
AI 中文摘要
我们研究了与有限群的幂半环相关的加法幂等半环簇格中的区间。我们证明了，位于阶至少为三的有限群的非空幂半环之上、且由有限多个有限群的完全幂半环生成的簇之下的每一个簇，都是非有限基的。事实上，这些簇中没有一个允许具有固定变量数有限界的恒等式基。特别地，添加空集给出一个完全由非有限基簇组成的区间。我们还构造了在零添加下封闭的等式上界，并由此得到具有相同性质的进一步区间。证明结合了有限交换商半环、有限模上核阻塞子的基数估计，以及有限群中有序积纤维的一致估计。作为推论，有限群的完全幂半环是有限基的当且仅当该群至多有两个元素。
英文摘要
We study intervals in the lattice of additively idempotent semiring varieties associated with power semirings of finite groups. We prove that every variety lying above the nonempty power semiring of a finite group of order at least three and below a variety generated by finitely many full power semirings of finite groups is nonfinitely based. In fact, none of these varieties admits an identity basis with a fixed finite bound on the number of variables. In particular, adjoining the empty set gives an interval consisting entirely of nonfinitely based varieties. We also construct equational upper bounds that are closed under zero adjunction and yield further intervals with the same property. The proof combines finite commutative quotient semirings, a cardinality estimate for kernel blockers over finite modules, and a uniform estimate for fibres of ordered products in finite groups. As a consequence, the full power semiring of a finite group is finitely based precisely when the group has at most two elements.
