What was already known
Every algebraic lattice is representable by an algebra, but its carrier may be infinite. Finite-lattice representation by a finite algebra remained a separate problem, and embedding a lattice into a partition lattice does not supply a full congruence representation.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Characterizes finite-algebra congruence lattices by a finite combinatorial witness.
Limitations
The linked Lean scope checks this colored-graph equivalence. The manuscript’s undecidability, nonrepresentable examples, and subgroup-interval conclusions are outside the selected formal statement.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A finite nonempty lattice is the full congruence lattice of some finite algebra exactly when it admits the finite colored-graph witness specified in the manuscript, whose edge colors encode the required congruence relations.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and precise partial or mismatched Lean-scope review for non-notifying local inclusion.
Read this published revision