发表机构
Technische Universität Ilmenau（伊尔梅瑙工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文反驳了Amarilli等人提出的非消去交集猜想，通过构造特定有限格作为反例，证明其普遍不成立，且所需素数$p$无需极大。
AI 中文摘要
Amarilli、Monet和Suciu提出的非消去交集（NCI）猜想[arXiv:2401.16210]指出，有限集合族的并集总能仅通过不交并和子集补，由其代数上非消去的交集构造而成。Wilhelm[arXiv:2608.19414]已证明，当要求见证的点代数表达式为左线性时，该猜想不成立；本文移除该限制，证明该猜想普遍为假：存在一个有限格，其顶元素完全没有点代数表示。反例是Wilhelm[arXiv:2608.19414]中的格$P_{p,\frak{m}}$，论证仅在两方面不同：其一，将Wilhelm的序贯“切换游戏”替换为对应树形对象平面树，平面树之于点代数树，恰如切换游戏之于左线性点代数树；其二，使用一个标记平面，其中不存在大小介于$2p$和$4p$之间的容许集，这也消除了对Erdős–Beck定理和算术Nullstellensatz的需求，因此$p$无需极大：每个素数$p\be 10^5$都适用。
英文摘要
The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential "toggle game" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erdős--Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \ge 10^{5}$ works.
Commentsv2: added a "Note added" reporting an independent equivalent result by A. Walz and comparing the two arguments; added a worked example of a winning plane da-tree over $\mathbb{F}_3^2$ (new figure); renamed "plane tree" to "plane da-tree" throughout; minor corrections and rewording; title hyphenation fixed
