发表机构
School of Mathematics, Northwest University（西北大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出加法幂等半环本质上非有限基的充分条件，构造满足条件的无限平坦半环，并刻画Zimin极小性，证明相关簇局部有限且本质上非有限基。
AI 中文摘要
我们给出了加法幂等半环本质上非有限基的一个充分条件。即，如果其生成的簇是局部有限的，并且每个Zimin字在加法序中都是极小的，那么它不包含在任何有限基的局部有限簇中。对于每个正整数$n$，我们构造了一个无限生成的平坦半环，它满足所有具有此极小性性质的半环在至多$n$个变量中的所有恒等式。我们还通过可数平坦因子半环$\finf$在生成簇中的成员关系来刻画Zimin极小性。簇$\fV(\finf)$是局部有限且本质上非有限基的。
英文摘要
We establish a sufficient condition for an additively idempotent semiring to be inherently nonfinitely based: if its generated variety is locally finite and every Zimin word is minimal in the additive order, then it is contained in no finitely based locally finite variety. We characterize Zimin minimality by the membership of a countable flat factor semiring $\Finf$ in the generated variety and prove that $\V(\Finf)$ is a minimal inherently nonfinitely based variety. We also obtain general restrictions on the additive order of finite inherently nonfinitely based ai-semirings. As applications, the six-element ai-semirings $\A$ and $\B$ are shown to be inherently nonfinitely based. Finally, we prove that the six-element ai semiring $\Abar$ is not inherently nonfinitely based and is finitely based. Its multiplicative reduct is inherently nonfinitely based; thus, to the best of our knowledge, $\Abar$ is the first finitely based ai-semiring with an inherently nonfinitely based multiplicative reduct.
