What does this research mean for the field?

Boolean antichains of any size in any finite lattice can be completely classified, yielding an exact finite formula for their enumerator and establishing a natural bijection between maximum Boolean antichains in a matroid's flat lattice and the bases of its simplification. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Classify and enumerate Boolean antichains of all sizes across arbitrary finite lattices, resolving open structural conjectures on geometric lattices and graphic matroids.

September 2, 2026Open Access

Boolean Antichains in Finite Lattices: Complete Classification, Product Structure, and Matroid Applications

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Authors

ALAlex Chengyu Li

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Implication

Mathematical analysis classifies Boolean antichains across finite lattices, revealing exact enumerators and direct bijections with matroid bases.

Key Points

Classify and enumerate Boolean antichains of all sizes across arbitrary finite lattices, resolving open structural conjectures on geometric lattices and graphic matroids.
Characterized Boolean families by intersecting subfamilies to generate candidate Boolean atoms and testing for vanishing reconstruction height gaps.
Derived enumerator formulas through binomial transforms under lattice products and verified theorem proofs using the Lean 4 proof assistant.
Proved that maximum Boolean antichains in the flat lattice of a rank-r matroid are in natural bijection with the bases of its simplification, verifying the spanning-tree bijection for graphic matroids.
Established an exact enumerator formula whose binomial transform is multiplicative under lattice products, generating explicit positive convolutions.
Obtained complete classifications and distributions for uniform matroids, finite distributive lattices, subspace lattices, and finite projective planes.

Cite This Study

Alex Chengyu Li (2026) studied this question.

synapsesocial.com/papers/6a97e305c562ede874ec795d https://doi.org/10.5281/zenodo.22201394

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Implication

Mathematical analysis classifies Boolean antichains across finite lattices, revealing exact enumerators and direct bijections with matroid bases.

Boolean Antichains in Finite Lattices: Complete Classification, Product Structure, and Matroid Applications

Discussion

Member takes

Implication

Key Points

Cite This Study

Boolean Antichains in Finite Lattices: Complete Classification, Product Structure, and Matroid Applications

Discussion

Member takes

Implication

Key Points

Cite This Study

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