平半环$S(W)$的有限基问题
The finite basis problem for the flat semirings $S(W)$
AI总结：
该研究探讨平半环$S(W)$的有限基问题，明确其有限基性与词集合$W$的长度、不含的幂次条件相关，为Jackson等人提出的开放问题提供了部分解答。
AI中文摘要：
我们聚焦于形如$S(W)$的平半环的有限基问题，其中$W$是任意非空词集合。我们证明：当$W$中每个词的长度至多为3时，$S(W)$生成一个Cross簇（因此是有限基的）；而当存在$k\geq3$使得$W$不含$x^{k+2}$但含$x^{k+1}$时，$S(W)$是非有限基的。特别地，若$W_k$表示所有长度为$k$的词的集合，则$S(W_k)$是有限基的当且仅当$k\leq3$。此外，当$W$有限且不含$x^4$时，$S(W)$是非有限基的。这些结果为Jackson等人（《代数学杂志》611卷：211-245，2022年）提出的一个开放问题提供了部分解答。
英文摘要：
We focus on the finite basis problem for flat semirings of the form $S(W)$, where $W$ is an arbitrary set of nonempty words. We prove that $S(W)$ generates a Cross variety (and hence is finitely based) whenever every word in $W$ has length at most $3$, whereas it is nonfinitely based whenever there exists $k \geq 3$ such that $W$ is $x^{k+2}$-free but not $x^{k+1}$-free. In particular, if $W_k$ denotes the set of all words of length $k$, then $S(W_k)$ is finitely based if and only if $k \leq 3$. Moreover, $S(W)$ is nonfinitely based whenever $W$ is finite and not $x^4$-free. These results provide a partial answer to an open problem raised by Jackson et al.~(J Algebra 611: 211--245, 2022).
