Finite lattice representation problem: is every finite lattice a congruence lattice of a finite algebra?
Statement
Is every finite lattice isomorphic to the congruence lattice $\mathrm{Con}(A)$ of some finite algebra $A$? By the Pálfy–Pudlák theorem (1980) this is equivalent to: does every finite lattice occur as an interval $[H,G]$ in the subgroup lattice of some finite group $G$? Resolve the problem — it is widely believed to have a negative answer, and effort is concentrated on finding a finite lattice (in particular of height $2$) that is NOT such a congruence lattice / subgroup-lattice interval.
Acceptance. ADVANCES: for specific small lattices, either an explicit realization as $[H,G]$ in a finite group (certificate: $G$, $H$, and the verified interval), extending the catalogue of known-representable lattices; or a rigorous non-representability argument for a candidate height-$2$ lattice. FULLY RESOLVES: a finite lattice proven not to be the congruence lattice of any finite algebra (equivalently, not an interval in any finite subgroup lattice), or a proof that all finite lattices are representable. Provide the group-theoretic certificate or the obstruction proof.
Background
One of the oldest open problems in universal algebra (roots in the Grätzer–Schmidt era, 1960s). Pálfy–Pudlák (1980) reduced it to the finite-group interval problem. Known: every finite DISTRIBUTIVE lattice is the congruence lattice of a finite lattice (classical, Funayama–Nakayama / Dilworth); many specific small lattices have been realized as intervals $[H,G]$ via explicit group actions. The search is for one small non-representable lattice, with height-$2$ lattices ($M_n$-type; the candidate sometimes denoted $L_7$) the focus. Still open as of 2026 (Wikipedia 'Finite lattice representation problem', updated March 2026; Open Problem Garden). Deep links to primitive permutation groups and coset-geometry / interval-lattice theory (see arXiv:1204.4305). An attacker must bring: for a concrete small candidate lattice $L$, a GAP-driven search attempting to realize $L$ as an interval $[H,G]$ across the finite-group catalogue (a positive realization is a finite certificate: the pair $(H,G)$), together with coset-geometry / O'Nan–Scott obstruction arguments that would prove a given $L$ non-representable. Decomposable lattice-by-lattice; a single certified non-representable finite lattice resolves the whole problem.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Finite lattice representation problem (Wikipedia, still open 2026) | website |
| REF-02 | Open Problem Garden — Finite lattice representation problem | website |
| REF-03 | Congruence lattices of finite algebras | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.