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)
Manuscripts (2)
Finite congruence lattices: characterization and undecidability
September 24, 2026 · source · cite as \cite{OAI:Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026}
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.
A negative solution to the finite lattice representation problem
September 24, 2026 · source · cite as \cite{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026}
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.
Result 206 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md