AI 中文总结
本文为八元素上三角布尔矩阵半环给出显式有限恒等式基，并刻画了关联半环的等式理论，核心贡献是结合乘法基与有限规则完成推导。
AI 中文摘要
本文给出了签名 $(+,\cdot)$ 下上三角布尔 $2\times2$ 矩阵构成的八元素半环的一个显式有限恒等式基。该基由已知的乘法基、ai-半环律以及 30 条混合恒等式组成，每条混合恒等式至多使用八个变量。证明将乘法还原的有限基与有限规则相结合，这些规则用于洗牌单词、复制标记出现以及交换相邻出现并添加指定见证。这将对单字母间隙准则的检验转化为对每个有效半环恒等式的推导。我们还研究了带子簇、子簇格中一个显著的 156 元素区间、同余、平坦成员以及自由代数的有限表示。对于 $m\geq2$，每个 $m$ 字母单词都有一个长度至多 $m^2$ 的等价子词，且自由代数具有双指数秩增长。对于任意偏序集，我们确定了非平凡有界分配格上的布尔关系半环和有限支撑关联半环的等式理论：有限高度 $h$ 给出 $T_h$ 的理论，而无界高度则恰好给出所有加法幂等半环的恒等式。
英文摘要
An explicit finite identity basis is given for the eight-element semiring of upper triangular Boolean \(2\times2\) matrices in the signature \((+,\cdot)\). The basis consists of a known multiplicative basis, the ai-semiring laws, and 30 mixed identities, each using at most eight variables. The proof combines a finite basis for the multiplicative reduct with finite rules for shuffling words, duplicating a marked occurrence, and interchanging adjacent occurrences while adding prescribed witnesses. This converts the one-letter gap criterion into a derivation of every valid semiring identity. We also study the band subvariety, a distinguished 156-element interval of the subvariety lattice, congruences, flat members, and finite representations of free algebras. Every \(m\)-letter word has an equivalent subword of length at most \(m^2\) for \(m\geq2\), and the free algebras have doubly exponential rank growth. For arbitrary partially ordered sets, we determine the equational theory of Boolean relation semirings and of finite-support incidence semirings over nontrivial bounded distributive lattices: finite height \(h\) gives the theory of \(T_h\), while unbounded height gives precisely the identities of all additively idempotent semirings.
CommentsThis paper is withdrawn. The manuscript is incomplete and requires further work
