$S_7$ 上的矩阵簇：三维稳定性与连续基数大小的子簇区间
Matrix varieties over $S_7$: dimension-three stability and continuum-sized subvariety intervals
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中文总结 AI 辅助
本文研究三元平坦加法幂等半环上的矩阵簇，完全确定其簇链，证明三维及以上生成同一稳定簇，并揭示连续基数大小的子簇区间及通用矩阵算子。
中文摘要 AI 辅助
我们研究三元平坦加法幂等半环上的矩阵半环，该半环的元素为一、a 和无穷，其中 a 的平方为无穷。我们完全确定了相关的矩阵簇链。标量、二阶和三阶情形生成三个不同的簇，而每个至少三阶的矩阵维度生成相同的簇。证明表明，任意维度中恒等式的任何失效都已在三个指标上被观察到，并且它为稳定簇中的所有恒等式提供了一个坐标判据。我们还实现了每个由入度和出度至多为一的有向图产生的图半环，作为二阶矩阵半环的直接幂的除数。因此，由所有三幂零平坦半环生成的簇包含在二阶矩阵簇中。然后，有向环和独立反转恒等式将奇素数的幂集格嵌入到基础簇与非平凡矩阵簇之间的每个区间中。因此，每个这样的区间具有连续基数大小，并包含连续基数大小的链和反链。最后，我们确定了通过删除常数全一矩阵而得到的乘法子半环的最后三个幂，并在加法幂等半环簇的格上发展了通用矩阵算子，包括稳定闭包、稳定核以及沿矩阵维度链的等式传播。
英文摘要
We study matrix semirings over the three-element flat additively idempotent semiring with elements one, a, and infinity, where the square of a is infinity. We determine the associated matrix-variety chain completely. The scalar, two-by-two, and three-by-three cases generate three distinct varieties, while every matrix dimension at least three generates the same variety. The proof shows that every failure of an identity in arbitrary dimension is already witnessed on three indices, and it yields a coordinate criterion for all identities in the stable variety. We also realize every graph semiring arising from a directed graph of in-degree and out-degree at most one as a divisor of a direct power of the two-by-two matrix semiring. Consequently, the variety generated by all three-nilpotent flat semirings is contained in the two-by-two matrix variety. Directed cycles and independent reversal identities then embed the power-set lattice of the odd primes into each interval between the base variety and a nontrivial matrix variety. Thus every such interval has continuum cardinality and contains continuum-sized chains and antichains. Finally, we determine the last three powers of the multiplicative subsemiring obtained by deleting the constant all-one matrix, and we develop general matrix operators on the lattice of additively idempotent semiring varieties, including stable closures, stable cores, and propagation of equality along matrix-dimension chains.
