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Maximal Sublattices of Convex Geometries with Unbounded Complement Minima

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Y. T. Miah ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

For every integer r≥3r\ge 3, we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly rr minimal elements and exactly two maximal elements. The lattice has (r+2)2(r+2)^2 elements and the sublattice has (r+1)(r+2)/2+3(r+1)(r+2)/2+3 elements. At r=3r=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.

Provenance statement

Y. T. Miah selected the problem and directed the investigation. OpenAI's GPT-6 Pro contributed substantially to developing the parametrised construction and mathematical arguments, investigating the literature, checking finite cases, and drafting the manuscript. The author reviewed the results and approved the final version. The work was first released on Zenodo on 25 October 2026 (DOI: 10.5281/zenodo.23632920).

Tools used

OpenAI
ChatGPTVersion 6 Pro

References

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Version history

  1. v1Submitted by Y. T. MiahInitial depositCurrentOct 25, 2026