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
本文研究偏序集区间半环的等式理论，刻画了由平坦半环生成的ai-半环簇的子簇结构，给出了有限恒等式基，并证明了每个真子簇包含于某个有限生成子簇，其中B_3为Cross簇且具有11个子簇。
AI 中文摘要
我们研究了由所有平坦半环 $S(a_1\cdots a_k)$（其中字母 $a_i$ 两两不同）生成的 ai-半环簇 $\V_\infty$ 的等式理论和子簇结构。利用偏序集的区间半环，我们刻画了其次直不可约成员，并借助严格保序映射的联合分离族描述了簇的成员关系。我们为 $\V_\infty$ 以及由 $S(a_1\cdots a_k)$ 生成的每个 $\V_k$ 得到了显式的有限恒等式基。因此，与非空线性词集合 $W$ 相关的每个平坦半环 $S(W)$ 都是有限基的。这为每个 $k\geq 1$ 提供了具有恰好 $k$-幂零乘法还原的有限基 ai-半环。对于每个 $k\geq 1$，令 $\B_k$ 为 $\V_\infty$ 中由 $(k+1)$-幂零恒等式定义的子簇。我们证明每个 $\B_k$ 都由一个有限区间半环生成，并且 $\V_\infty$ 的每个真子簇都包含在某个 $\B_k$ 中。簇 $\B_3$ 是一个具有恰好 $11$ 个子簇的 Cross 簇，而 $[\V_k,\B_k]$、$[\V_k,\V_{k+1}]$ 和 $[\B_{k-1},\B_k]$ 对于每个 $k\geq 4$ 都包含连续统多个子簇。特别地，这为每个 $k\geq 5$ 提供了无穷多个有限基的有限半环 $S(a_1\cdots a_k)$，其生成的簇具有连续统多个子簇。
英文摘要
We study the equational theory and subvariety structure of the ai-semiring variety $\V_\infty$ generated by all flat semirings $S(a_1\cdots a_k)$, where the letters \(a_i\) are pairwise distinct. Using interval semirings of posets, we characterize its subdirectly irreducible members and describe variety membership in terms of jointly separating families of strict order-preserving maps. We obtain explicit finite identity bases for \(\V_\infty\) and each $\V_k$ generated by $S(a_1\cdots a_k)$. Consequently, every flat semiring \(S(W)\) associated with a nonempty set \(W\) of linear words is finitely based. This yields finitely based ai-semirings with exactly \(k\)-nilpotent multiplicative reduct for each \(k\geq 1\). For each \(k\geq 1\), let \(\B_k\) be the subvariety of \(\V_\infty\) defined by the \((k+1)\)-nilpotent identity. We prove that each \(\B_k\) is generated by a finite interval semiring and that every proper subvariety of \(\V_\infty\) is contained in some \(\B_k\). The variety \(\B_3\) is a Cross variety with exactly \(11\) subvarieties, whereas \([\V_k,\B_k]\), \([\V_k,\V_{k+1}]\), and \([\B_{k-1},\B_k]\) each contain continuum many subvarieties for every $k\geq 4$. In particular, this provides infinitely many finitely based finite semirings $S(a_1\cdots a_k)$ whose generated variety has continuum many subvarieties for every \(k\geq 5\).
Comments35 pages
