Refutation of the Non-Cancelling Intersections Conjecture
Abstract
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 Pp,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 ≥ 105 works.
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BibTeX
@misc{wilhelm2026refutation,
title = {{Refutation of the Non-Cancelling Intersections Conjecture}},
author = {Hermann Wilhelm},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27416},
eprint = {2608.27416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27416},
}
RIS
TY - UNPB
TI - Refutation of the Non-Cancelling Intersections Conjecture
AU - Hermann Wilhelm
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.27416
UR - https://arxiv.org/abs/2608.27416
AB - 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,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 ≥ 10^5 works.
ER -