HEXAGONbeta

Maximal sublattices of finite semidistributive lattices: the complement need not be an interval

A mix of human-written and AI-generated textHuman understanding: some partsmath.RA — Rings and Algebras

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 23, 2026 / CC BY 4.0

Abstract

Adams, Freese, Nation and Schmid proved that the complement of a maximal sublattice of a finite bounded lattice is an interval, and asked whether the same holds for finite semidistributive lattices; Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that it does. It does not. In a 26-element semidistributive sublattice of the square of a 14-element semidistributive lattice, the diagonal is a maximal sublattice whose complement has two minimal elements.

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. Refutes Hypothesis 1 of Adaricheva et al. 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/Conjecture-001/releases/tag/v1.0.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 1 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

  1. M. E. Adams, R. Freese, J. B. Nation and J. Schmid, Maximal sublattices and Frattini sublattices of bounded lattices , J. Austral. Math. Soc. Ser. A 63 (1997), 110--127. 10.1017/S1446788700000355DOI
  2. K. Adaricheva, A. Mata, S. Silberger and A. Zamojska-Dzienio, Conjecture on maximal sublattices of finite semidistributive lattices and beyond , Discrete Math. Theor. Comput. Sci. 28 :2 (2026). 10.46298/dmtcs.16374DOI
  3. D. Bhattacharjee, Maximal sublattices of finite semidistributive lattices: paper, Lean 4 development and search programs , Zenodo, 2026. 10.5281/zenodo.23353448 ; https://github.com/creelie/Conjecture-001DOI
  4. D. Bhattacharjee, Maximal sublattices of convex geometries: four counterexamples , Zenodo, 2026. 10.5281/zenodo.23343145DOI
  5. C. C. Chen, K. M. Koh and S. K. Tan, Frattini sublattices of distributive lattices , Algebra Universalis 3 (1973), 294--303. 10.1007/BF02945131DOI
  6. A. Day, Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices , Canad. J. Math. 31 (1979), 69--78. 10.4153/CJM-1979-008-xDOI
  7. R. Freese, J. Ježek and J. B. Nation, Free Lattices , Mathematical Surveys and Monographs 42 , American Mathematical Society, Providence, RI, 1995. 10.1090/surv/042DOI
  8. The mathlib Community, The Lean mathematical library , in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), ACM, New York, 2020, 367--381. 10.1145/3372885.3373824DOI
  9. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci. 12699 , Springer, Cham, 2021, 625--635. 10.1007/978-3-030-79876-5_37DOI
  10. I. Rival, Maximal sublattices of finite distributive lattices , Proc. Amer. Math. Soc. 37 (1973), 417--420. 10.1090/S0002-9939-1973-0311527-9DOI

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 23, 2026