线性字的平坦半环的有限恒等基
Finite identity bases for flat semirings of linear words
- Chongqing University of Technology（重庆理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明线性字集合的平坦半环总有有限恒等基，给出显式基，并以十一元半环$S(abcd)$反例推翻长度界猜想。
AI中文摘要：
对于非空字的集合$W$，令$S(W)$为由$W$中字的非空因子连同吸收零元构成的平坦半环。我们证明，当$W$中的每个字都是线性字时，$S(W)$具有有限恒等基，且对$W$的大小或其字的长度没有任何限制。在有界情形下，其簇由区间半环$A_m\cong S(a_1\cdots a_m)$生成，其中$m$是最大字长，字母$a_i$互不相同。在无界情形下，其簇由非负整数所有有限区间上的区间半环生成。我们在两种情形下都给出了显式的有限基。证明通过端点图对非零多项式赋值进行编码，并利用有限多个拼接恒等式推导出所需的图等同关系。特别地，十一元半环$S(abcd)$具有有限基，这为Gao、Ren和Zhao提出的关于$S(W)$的长度界猜想提供了反例。
英文摘要：
For a set $W$ of nonempty words, let $S(W)$ be the flat semiring formed by the nonempty factors of words in $W$, together with an absorbing zero. We prove that $S(W)$ has a finite identity basis whenever every word in $W$ is linear, with no restriction on the size of $W$ or on the lengths of its words. In the bounded case its variety is generated by the interval semiring $A_m\cong S(a_1\cdots a_m)$, where $m$ is the maximum word length and the letters $a_i$ are distinct. In the unbounded case its variety is generated by the interval semiring on all finite intervals of the nonnegative integers. We give explicit finite bases in both cases. The proofs encode nonzero polynomial evaluations by endpoint graphs and derive the required graph identifications using finitely many splicing identities. In particular, the eleven-element semiring $S(abcd)$ is finitely based, providing a counterexample to the length-bound conjectures for $S(W)$ proposed by Gao, Ren and Zhao.
