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Math-2335 · mathematics · Group theory

Pálfy–Pudlák Finite-Lattice Representation Problem

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every nonempty finite lattice $\Lambda$, do there exist a finite group $G$ and $H\leq G$ such that $\Lambda$ is isomorphic to the inclusion-ordered lattice of all subgroups between $H$ and $G$?

Formulation references: [S1], [S2].

Short English statement

For every nonempty finite lattice $\Lambda$, do there exist a finite group $G$ and $H\leq G$ such that $\Lambda$ is isomorphic to the inclusion-ordered lattice of all subgroups between $H$ and $G$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2335-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2335

Theorem1.2 supplies a finite lattice excluded from every full subgroup interval of every finite group, precisely negating the original universal interval-representation statement. Theorem1.1(iv) makes this representability predicate undecidable. If every lattice were representable it would have the constant-yes algorithm, contradiction; hence the universal original question is refuted. Within these assumptions, this is a claimed negative answer to the stated question.

Pálfy–Pudlák Finite-Lattice Representation Problem

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Some finite lattice is claimed to have no realization as the complete congruence lattice of any finite algebra; A colored-graph criterion is claimed to characterize realizable finite congruence lattices, with an undecidable recognition problem; recognition of finite subgroup intervals is also claimed to be undecidable. Together they give a claimed negative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
The two existence claims concern finite lattices. First, some nonempty finite lattice $L$ satisfies \[ \forall A\text{ finite nonempty algebra of finite signature},\qquad L\not\cong\operatorname{Con}(A). \] Second, there is a finite lattice $L'$ for which \[ \forall G\text{ finite},\ \forall D\le G,\qquad L'\not\cong[D,G]. \] Here $\operatorname{Con}(A)$ is the congruence lattice and $[D,G]$ is the complete interval of subgroups containing $D$.
Result 2.
For a nonempty finite lattice $L$, let $\mathcal R(L)$ mean that $L\cong\operatorname{Con}(A)$ for some finite algebra $A$. The claimed finite colored-graph criterion is equivalent to $\mathcal R(L)$, with a finite decision procedure once the candidate graph size is fixed. Nevertheless, \[ \text{no total algorithm decides }\mathcal R(L)\text{ from the order table of }L. \] The source also specifies a nonrepresentable lattice $L_0$ of least cardinality, with $|L_0|>7$ and every smaller finite lattice representable. Its group-interval assertion is \[ \text{no total algorithm decides }\exists G\text{ finite},\ H\le G:\ L\cong[H,G]. \] The undecidability theorem additionally rules out a total computable bound, depending only on $|L|$, on the smallest carrier size among representable lattices.

Outline of the argument

The obstruction is first phrased for subgroup intervals using specially tested lattice chains and coordinate transport. Classification-based reductions control possible factor actions, while competing rank and growth estimates contradict the required long configuration; the subgroup-interval equivalence transfers the obstruction to finite algebras.

Finite colored graphs encode the equivalences permitted by all operations of an algebra. Small witness configurations are assembled into lattice representation constraints, and arithmetic divisor obstructions rule out bounded algebraic realizations; a reduction turns the characterization into the claimed undecidability result.

Formal verification

For family 206, the documented theorem identifies representability with a finite colored-graph witness. Undecidability, nonrepresentation, and subgroup-interval conclusions are not contained in that selected graph equivalence. The listed Comparator files are FiniteCongruenceGraph.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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