所有三阶半环生成的簇不是有限基的
The variety generated by all semirings of order three is nonfinitely based
- 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：
本文证明所有三阶半环（及三阶加法幂等半环）生成的簇不是有限基的，通过锚定奇圈恒等式和排除变量数上界的方法确立结论。
AI中文摘要：
我们证明了所有三阶半环生成的簇不是有限基的，其中加法不要求交换且签名中不含常量。同样的结论也适用于所有三阶加法幂等半环生成的簇。我们通过排除恒等式基中变量数目的统一上界来确立这些结论。我们的证明使用了与锚定奇圈相关的恒等式。三个小的交换测试半环分离出一个由恰好两个多项式组成的多项式等价类。对于长度为$n$的圈，它们之间的每一个第一个非平凡推导都需要一个至少含有$n+1$个变量的恒等式，即使在多项式替换下也是如此。一个收缩后跟一个带商将吸收恒等式转移到任意加法上，并将完全联合簇的ai-子簇与ai-生成元的联合簇等同起来。圈恒等式的有效性来自对链加法和平坦加法的结构分析。一个关于阶至多为三的半群的基本六次幂引理提供了收缩。
英文摘要：
We prove that the variety generated by all semirings of order three is nonfinitely based, where addition is not required to be commutative and the signature has no constants. The same conclusion holds for the variety generated by all additively idempotent semirings of order three. We establish these conclusions by excluding a uniform bound on the number of variables in an identity basis. Our proof uses identities associated with anchored odd cycles. Three small commutative test semirings isolate a polynomial equivalence class consisting of exactly two polynomials. For a cycle of length $n$, every first nontrivial deduction between them requires an identity with at least $n+1$ variables, even under polynomial substitutions. A retraction followed by a band quotient transfers absorption identities to arbitrary addition and identifies the ai-subvariety of the full joint variety with the joint variety of the ai-generators. Validity of the cycle identities follows from a structural analysis of chain and flat addition. An elementary sixth-power lemma for semigroups of order at most three supplies the retraction.
