AI 中文总结
本文解决了六元交换半群上四个相容加法的有限基与子簇问题，给出了分类、无限基及子簇格结构，其中$\V(R_{12})$的子簇格可数无限。
AI 中文摘要
设 $M$ 为六元交换半群，它是半环 $SR_6$ 和 $TR_6$ 的公共乘法约简。Shao、Ren 和 Gao 的论文（参考文献 \cite{ShaoRenGao2026}）的结尾段落提出了关于 $M$ 上剩余四个相容加法的有限基和子簇问题。我们针对四个同构类型 $R_{01},R_{02},R_{11},R_{12}$ 回答了这些问题。首先，我们分类了 $M$ 上所有相容加法：共有九个标记加法和六个同构类型，由 $R_{ij}$ 参数化，其中 $0\leq i\leq j\leq 2$。对于四个新类型中的每一个，我们给出了每个恒等式的图论判据、一个显式无限基以及不可有限基的证明。生成的簇 $\V(R_{01})$ 和 $\V(R_{02})$ 各有十一个子簇，而 $\V(R_{11})$ 有六十六个子簇。格 $\Sub(\V(R_{12}))$ 是可数无限的。该簇中的每个恒等式可化简为二十五个固定恒等式以及两个单调路径族 $\gamma_n$ 和 $\gammaD_n$ 的子集。这产生了一个规范签名 $(H,p,q)$、完整范式、显式交与并运算以及所有覆盖的公式。共有 153 个固定节点、43 个单参数族和 9 个双参数族；恰好有十八个子簇是有限基的，唯一的极限子簇是 $\V(SR_6)$。四个有限半环的强非有限基状态仍然开放。
英文摘要
Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\leq i\leq j\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\V(R_{01})$ and $\V(R_{02})$ have eleven subvarieties each, while $\V(R_{11})$ has sixty-six. The lattice $\Sub(\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $γ_n$ and $\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.
