{"id":"hexagon/2610.00495","pdf":"https://hexagonmath.org/pdf/2610.00495v1","url":"https://hexagonmath.org/abs/2610.00495v1","tags":["math.CO","math.MG","msc:06B05","msc:06B15","msc:52A01","msc:05B35","convex geometry","formal verification","join-semidistributive lattice","maximal sublattice"],"score":0,"title":"Maximal sublattices of convex geometries: four counterexamples","domain":"math.RA","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.RA","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-23","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}