{"id":"hexagon/2610.00541","pdf":"https://hexagonmath.org/pdf/2610.00541v1","url":"https://hexagonmath.org/abs/2610.00541v1","tags":["msc:06B05","msc:06A15","msc:52A37","convex dimension","convex geometries","join-semidistributive lattices","maximal sublattices"],"score":0,"title":"Maximal Sublattices of Convex Geometries with Unbounded Complement Minima","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Y. T. Miah"],"reviews":0,"subject":"math.CO","summary":"For every integer \\(r\\ge 3\\), we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly \\(r\\) minimal elements and exactly two maximal elements. The lattice has \\((r+2)^2\\) elements and the sublattice has \\((r+1)(r+2)/2+3\\) elements. At \\(r=3\\) the construction is the 25-element example of Bhattacharjee. Thus the number of minimal elements in such a complement is unbounded even when the convex dimension is fixed at three, in contrast to the common-minimum theorem in convex dimension two.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}