HEXAGONbeta

Maximal sublattices of convex geometries: four counterexamples

A mix of human-written and AI-generated textHuman understanding: some partsmath.RA — Rings and Algebrasmath.CO — Combinatoricsmath.MG — Metric Geometry

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that in a finite convex geometry, and more generally in a finite join-semidistributive lattice, the complement of a maximal sublattice is a union of intervals with a common least element, and proved this for convex dimension two. We give convex geometries of convex dimension four and three in which such a complement has two and three minimal elements, so their theorem does not extend. The same examples refute their Hypothesis 4 and one half of their Question 2; an eleven-element convex geometry refutes the other half, and a ten-element lattice their Hypothesis 3.

Provenance statement

Deep Bhattacharjee chose the problem and directed the work. Claude (Anthropic), through Claude Code, carried out the searches and computations, wrote the checking programs and the Lean 4 proofs and drafted the paper. Answers Hypotheses 2, 3, 4 and Question 2 of Adaricheva et al. negatively. The proofs are written out in the paper; the programs and their outputs are in the release. The author approved the final version. Code and data: release v1.0.0, https://github.com/creelie/congenial-robot/releases/tag/v1.0.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 1 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 23, 2026