OpenAI · 206-01 · 2026-09-24 · Algebra
Finite congruence lattices: characterization and undecidability
preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026
Family 206. Finite lattice representation: counterexamples and undecidability.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.
We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 206: Finite lattice representation and undecidability. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.