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Math-2513 · mathematics · Algebra

Deciding finite-algebra congruence-lattice representability

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does an algorithm exist that, on every order table of a finite nonempty lattice L, halts and determines whether L is isomorphic to Con(A) for some finite algebra A with arbitrary finitary operations?

Short English statement

Does an algorithm exist that, on every order table of a finite nonempty lattice L, halts and determines whether L is isomorphic to Con(A) for some finite algebra A with arbitrary finitary operations?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2513-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2513

The cited theorem explicitly forbids any total algorithm deciding finite-algebra representability from finite lattice tables. The formalization in lean/docs/206.md is only the graph criterion; it explicitly excludes undecidability. Within these assumptions, this is a claimed negative answer to the stated question.

Deciding finite-algebra congruence-lattice representability

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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

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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