OpenAI · 206-02 · 2026-09-24 · Algebra
A negative solution to the finite lattice representation problem
preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-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 a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.
Preprint source and Lean scope
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A Lean scope note is stored for family 206, but it does not name this preprint. It names 206-01. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.