AI 中文总结
研究矩阵半环$\mathbf{M}_n(S_7)$的有限基问题，通过证明嵌入定理得到簇链，表明特定区间内簇非有限基，得出$\mathbf{M}_n(S_7)$非有限基及相关结论，还证明其乘法约化的幂零性。
AI 中文摘要
首先证明了加法幂等半环$S$上矩阵半环$\mathbf{M}_n(S)$的嵌入定理：对于所有$n\geq2$，$\mathbf{M}_n(S)$嵌入到$\mathbf{M}_{n + 1}(S)$。当$S$是二元分配格时，得到严格递增的簇链。接着表明区间$[\mathsf{V}(S_c(abc)), \mathsf{V}(\mathbf{M}_n(S_7))]$中的每个簇都是非有限基的，从而得出$\mathbf{M}_n(S_7)$是非有限基的，且该区间包含至少可数无穷多个不同簇。还证明了$\mathbf{M}_n(S_7)$去掉常数矩阵$[1]_n$后的乘法约化是5 - 幂零的。
英文摘要
We first prove that two matrix semirings $\mathbf{M}_n(S_1)$ and $\mathbf{M}_n(S_2)$ are equationally equivalent whenever additively idempotent semirings $S_1$ and $S_2$ are equationally equivalent. We then prove an embedding theorem for matrix semirings $\mathbf{M}_n(S)$ over an additively idempotent semiring $S$: for all $n \geq 2$, $\mathbf{M}_n(S)$ embeds into $\mathbf{M}_{n+1}(S)$. This yields an ascending chain of varieties $\mathsf{V}(\mathbf{M}_2(S)) \leq \mathsf{V}(\mathbf{M}_3(S)) \leq \cdots$, which is strictly ascending when $S$ is the two-element distributive lattice. Finally, we show that every variety in the interval $[\mathsf{V}(S_c(abc)), \mathsf{V}(\mathbf{M}_n(S_7))]$ is nonfinitely based (i.e., has no finite basis for its identities), where $S_c(abc)$ is an eight-element flat semiring and $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. Consequently, $\mathbf{M}_n(S_7)$ is nonfinitely based, yielding an ascending chain $\mathsf{V}(\mathbf{M}_2(S_7)) \leq \mathsf{V}(\mathbf{M}_3(S_7)) \leq \cdots$; moreover, every variety in $[\mathsf{V}(S_7), \mathsf{V}(\mathbf{M}_n(S_7))]$ is also nonfinitely based, and this interval contains at least countably infinitely many distinct varieties.
