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
本文给出六元加法幂等半环TR_6的无限等式基，证明其生成极限簇，并研究其与SR_6生成的簇的子簇结构。
AI 中文摘要
我们为六元加法幂等半环 $TR_6$ 给出了一个显式的无限等式基，并证明了 $TR_6$ 是非有限基的。我们还完整描述了由 $TR_6$ 生成的簇的子簇格，表明它形成一个四元链。我们的结果表明，由 $TR_6$ 生成的簇是一个极限簇：它本身是非有限基的，但其所有真子簇都是有限基的。这提供了一个新的加法幂等半环的极限簇，不同于所有先前已知的极限簇。事实上，$\mathsf{V}(TR_6)$ 是由最大加代数 $\mathbf{N}$ 生成的簇的第一个显式极限子簇。此外，$TR_6$ 不是强非有限基的：它属于由有限加法幂等半环生成的有限基簇。连同六元加法幂等半环 $SR_6$，它们是前两个非有限基但非强非有限基的有限加法幂等半环。最后，我们研究了由 $SR_6$ 和 $TR_6$ 生成的簇，证明它是非有限基的，并且恰好有九个子簇，其中四个是非有限基的，其余五个是有限基的。
英文摘要
We present an explicit infinite equational basis for the six-element additively idempotent semiring $TR_6$ and prove that $TR_6$ is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by $TR_6$, showing that it forms a four-element chain. Our results demonstrate that the variety generated by $TR_6$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, $\mathsf{V}(TR_6)$ is the first explicit limit subvariety of the variety generated by the max-plus algebra $\mathbf{N}$. Moreover, $TR_6$ is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring $SR_6$, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by $SR_6$ and $TR_6$, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.
