发表机构
School of Mathematics, Northwest University（西北大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究截断循环群平坦半环的有限基问题，证明其恰在m≤2或(m,n)=(3,1)时具有有限基，并给出显式基及区域外的反模型构造。
AI 中文摘要
对于正整数 \\(m,n\\)，令 \\(A_{m,n}=(\{1,\ldots,m\}\times\Z_n)\cup\{0\}\\) 具有平坦加法与在次数 \\(m\\) 处截断的乘法。我们证明 \\(A_{m,n}\\) 恰在 \\(m\le 2\\) 或 \\((m,n)=(3,1)\\) 时具有有限基。在整个该区域内给出了显式有限基。在该区域之外，具有常和刚性性质的高围长超图对等式理论的每个有界变量片段都产生有限反模型。证明对参数不施加整除性或互素性限制。
英文摘要
For positive integers \(m,n\), let \[ A_{m,n}=(\{1,\ldots,m\}\times\Z_n)\cup\{0\} \] have flat addition and multiplication truncated at degree \(m\). We prove that \(A_{m,n}\) is finitely based exactly when \(m\le 2\) or \((m,n)=(3,1)\). Explicit finite bases are supplied throughout this region. Outside it, high-girth hypergraphs with a constant-sum rigidity property yield finite countermodels to every bounded-variable fragment of the equational theory. The proof places no divisibility or coprimality restriction on the parameters.
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