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Math-628 · mathematics · Algebra

Finite Lattice Representation Problem

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Statement

Definitions and mathematical statement

Shared notation.
Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A finite algebra is a finite nonempty set equipped with finitary operations. A congruence is an equivalence relation preserved by every operation. The set $\operatorname{Con}(A)$ of congruences forms a lattice under inclusion, with meet given by intersection. Ask whether each finite lattice is isomorphic, as a lattice, to $\operatorname{Con}(A)$ for some finite algebra. [S1] [S2]

Statement recorded in the supplied source.
Is every finite lattice isomorphic to the congruence lattice of some finite algebra? [S1]
Residual task reported by the archive.

The embedded review (2026-08-17) records: Represent every finite lattice or prove a finite nonrepresentable lattice. [S1]

Short English statement

Can every finite lattice arise as the collection of operation-compatible equivalence relations on a finite algebra?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 4

Math-628-A4 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-628

Theorem 1.1 establishes exactly the record’s negative existence formulation, excluding every finite algebra rather than only one fixed variety or signature. Theorem 1.1(iii) establishes exactly the record’s negative existence formulation, excluding every finite algebra rather than only one fixed variety or signature. Within these assumptions, this is a claimed affirmative answer to the stated question.

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 affirmative 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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Math-628-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-628

Theorem 1.1 gives a finite lattice excluded from the full congruence lattice of every finite algebra, which disproves the record’s universal representation assertion. Theorem 1.1(iii) gives a finite lattice excluded from the full congruence lattice of every finite algebra, which disproves the record’s universal representation assertion. Within these assumptions, this is a claimed negative answer to the stated question.

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

Go to contribution →

Math-628-A3 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-628

Finite Lattice Representation Problem

Definitions and mathematical statement

A finite lattice is a finite nonempty partially ordered set $L$ in which every pair has a greatest lower bound $x\wedge y$ and a least upper bound $x\vee y$. An algebra $\mathbf A=(A;(f_i)_{i\in I})$ consists of a nonempty set $A$ together with specified operations $f_i:A^{n_i}\to A$ of finite arities $n_i$. Here "finite algebra" means that $A$ is finite; its operations may be chosen as part of a proposed representation.

A congruence of $\mathbf A$ is an equivalence relation $\theta$ on $A$ compatible with every operation: whenever $a_j\mathrel\theta b_j$ for all inputs, one has $f_i(a_1,\ldots,a_{n_i})\mathrel\theta f_i(b_1,\ldots,b_{n_i})$. The set $\operatorname{Con}(\mathbf A)$, ordered by inclusion, is a lattice. Its meet is intersection, and its join is the smallest congruence containing both relations. A representation requires an isomorphism of lattices, hence preservation of both meet and join, and not merely an embedding.

The catalogue conjecture is the negative form of the finite lattice representation problem: \[ \exists\text{ finite lattice }L\quad \forall\text{ finite algebras }\mathbf A,\qquad L\not\cong\operatorname{Con}(\mathbf A). \] Thus a positive resolution of this conjecture must exhibit a finite lattice and exclude every possible finite underlying set and collection of operations. Its negation says that every finite lattice admits such a representation; the size of the representing algebra is not bounded in advance. No fixed algebraic signature, such as groups or rings alone, is imposed.

Short English statement

Is there a finite lattice that cannot be the lattice of compatible equivalence relations of any finite algebra?

Research attempt

Scope and process.
This is a GPT-6 Astra Ultra research attempt dated 1 October 2026, using primary-source browsing and elementary mathematical checks. The canonical definition above is retained. Results below are restricted results or reductions, not a claim of a new solution to the entire catalogue question. No formal proof assistant or external mathematical referee checked this text.
Literature and target.
The unrestricted finite-algebra question is the one in [S2]. DeMeo's research monograph [R1] discusses its relationship with subgroup intervals; Chan [R2] explicitly records representability of every finite distributive lattice. We give a direct construction of that known restricted result, with a bound on the size of the underlying set. No assertion about a particular smallest unknown lattice is needed here.
Plan.
A failed search within groups alone cannot exclude arbitrary algebras. Instead, choose operations to force the congruences to be coordinate subspaces, then encode an arbitrary finite partial order by additional operations. This tests which obstruction a proposed negative example would have to escape.
A finite algebra for any lattice of order ideals.
Let $P$ be a finite partially ordered set and let $V=\mathbb F_2^P$. Equip $V$ with addition, the constant zero, every coordinate projection $\pi_p(x)=x_pe_p$, and, for every $q\le p$, the unary map $T_{qp}(x)=x_pe_q$. These are finitely many operations on $2^{|P|}$ elements. For a congruence $\theta$, put $W=\{x:x\mathrel\theta0\}$. Compatibility with addition shows that $W$ is a subspace and \[ x\mathrel\theta y\quad\Longleftrightarrow\quad x+y\in W. \] Indeed, add $y$ to both sides for one implication and add $y$ back for the converse. Conversely, a subspace stable under all the stated linear maps defines a compatible equivalence relation by this formula.

Projection stability implies $W=\operatorname{span}\{e_p:p\in I\}$ for some $I\subseteq P$. To see this, every coordinate present in a vector of $W$ can be isolated by $\pi_p$, and every vector is the sum of its coordinate projections. The maps $T_{qp}$ then imply $p\in I,\ q\le p\Rightarrow q\in I$. Conversely any such down-set is stable under all operations. Thus \[ \operatorname{Con}(V;+,0,\pi_p,T_{qp})\cong J(P), \] where $J(P)$ is the lattice of down-sets. Inclusion, intersection and union correspond exactly; this is an isomorphism, not just an embedding.

Recovering all finite distributive lattices.
For completeness, let $L$ be finite distributive and let $P$ be its nonzero join-irreducible elements. Every $x\in L$ is the join of the members of $P$ below it, by induction on the finite order. Moreover a join-irreducible $p$ is join-prime: if $p\le a\vee b$, then $p=(p\wedge a)\vee(p\wedge b)$, so irreducibility gives $p\le a$ or $p\le b$. Consequently \[ x\longmapsto\{p\in P:p\le x\} \] preserves meets and joins and is injective. A down-set $I$ is its image at $x=\bigvee I$: join-primality ensures that no extra join-irreducible lies below this join. This proves surjectivity, including the empty down-set. Composing with the construction proves that $L$ has a finite representation on at most $2^{|P|}$ elements.
Verification and obstruction to extending the argument.
If $P$ is an antichain, only coordinate restrictions remain, yielding a Boolean lattice. If $P$ is a chain, the possible subspaces are its initial coordinate spans, yielding a chain of congruences. For $P$ empty the one-element algebra gives the one-element lattice. These checks include the extreme cases in the statement. The crucial join-primality step uses distributivity: in the five-element diamond a join-irreducible atom lies below the join of the other two atoms without lying below either one. Our operations always give a distributive ideal lattice, so this particular construction cannot represent that diamond. This is an obstruction to the construction, not to all finite algebras.
Final status and first remaining gap.
UNRESOLVED. Every finite distributive candidate is excluded from being a counterexample by an explicit finite representation. There is no argument here excluding a nondistributive lattice from all finite algebras, nor a representation theorem for every nondistributive lattice. Those are the missing alternatives required by the target.

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