Result 206, Algebra

Finite lattice representation and undecidability

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Finite lattices describe ordered patterns of relationships. These manuscripts claim that some such patterns cannot arise as the complete system of operation-compatible equivalences in any finite algebra, and that recognizing representable patterns is undecidable.

What changes?

A lattice is an ordered structure where every pair has a greatest common lower bound and a least common upper bound. Congruences are equivalence relations preserved by an algebra's operations; ordered by inclusion, they form its congruence lattice. The unreviewed manuscripts report that some finite nonempty lattices cannot be the full congruence lattice of any finite nonempty algebra with finitely many operations. They also claim that no algorithm can decide this representability for every finite nonempty input lattice.

What does that help mathematicians do?

The reported colored-graph characterization gives an exact structural criterion for representability, but not a universal decision procedure. This distinction matters: researchers could use the criterion to study individual lattices, while undecidability rules out a test guaranteed to settle every case. The manuscript also reports undecidable recognition of full subgroup intervals in finite groups: lattices consisting of all subgroups between two fixed subgroups, ordered by inclusion.

Are there practical applications?

Its immediate value is foundational: it identifies limits on modeling finite order structures through algebraic equivalences and finite groups. For automated algebra, the undecidability claims impose a precise boundary: procedures may handle restricted families or particular examples, but cannot be guaranteed to decide these representation questions for every finite lattice.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Finite congruence lattices: characterization and undecidability

September 24, 2026 180 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026,
  author = {{OpenAI}},
  title = {{Finite congruence lattices: characterization and undecidability}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf}{OAI:Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026}},
  year = {2026}
}

A negative solution to the finite lattice representation problem

September 24, 2026 87 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026,
  author = {{OpenAI}},
  title = {{A negative solution to the finite lattice representation problem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf}{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/206.md.

Finite lattice representation and undecidability

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The finite lattice representation problem asks which finite lattices occur as the full congruence lattice of a finite algebra. The formalization proves the paper's colored-graph characterization: a finite nonempty lattice is representable exactly when it admits the specified finite nonempty graph witness, whose edge colors encode the required congruence relations. This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it.

Comparator links

Result Comparator statement
Colored-graph criterion for finite congruence lattices FiniteCongruenceGraph.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.