Problem dossier · Algebra
Finite lattice representation problem
Is every finite lattice isomorphic to the congruence lattice of a finite algebra?
§1
Status
Open. Equivalent (Pálfy–Pudlák) to a question about intervals in subgroup lattices of finite groups; known for many lattice classes.
Think you can crack this one? Read the playbook before you announce →
§2
The Angle of Attack
Enumerate small lattices and search for representing algebras/groups computationally; focus on the smallest lattices with no known representation.
Tags: lattices · universal algebra · congruences · groups
§3
The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
§4
The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
§5
Related Problems
More open problems in Algebra and adjacent territory.
- Birch–Tate conjecture — Algebra
- Green's conjecture on syzygies — Algebra
- Rota's basis conjecture — Algebra
- Williamson matrices — Algebra
- Kissing numbers in higher dimensions — Geometry & Packing