---
tag: W83D78WN7T
kind: problem
name: Finite lattice representation problem
msc: ["06B15", "08A30", "20E15"]
uses: [WH4SFAACAS, REE07CSXHR, 0B8BW4YWTF, QJGPT2W0FT, DCSXXT4SER]
about: [REE07CSXHR, 0B8BW4YWTF]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_of_subgroups
history:
  posed_by: "Péter Pál Pálfy and Pavel Pudlák"
  first_stated: "1980"
  notes: "Pálfy and Pudlák stated the problem in its form about intervals in subgroup lattices of finite groups. The question for finite algebras was open since Grätzer and Schmidt showed in 1963 that every algebraic lattice is the congruence lattice of some algebra."
---

## Problem

Let $L$ be a finite [[lattice|WH4SFAACAS]].
Is there always a finite [[algebra|QJGPT2W0FT]] $A$ whose [[congruence lattice|REE07CSXHR]] is isomorphic to $L$?

## Remarks

Pálfy and Pudlák proved in 1980 that the problem has an equivalent form in group theory.
The problem has a positive answer if and only if the following holds.
Every finite lattice is isomorphic to an interval $[H, G] = \set{K : H \le K \le G}$ in the [[lattice of subgroups|0B8BW4YWTF]] of some finite group $G$.
Here $H$ is a subgroup of $G$, and the interval is ordered by inclusion.
The problem is open.

Without the finiteness condition on $A$, the answer is yes.
By a theorem of Grätzer and Schmidt, every algebraic lattice, and so every finite lattice, is the congruence lattice of some algebra.
The difficulty is to find a finite algebra.
For a finite distributive lattice the answer is also yes, since it is the congruence lattice of a finite lattice.

## Explanations

### L1 human

Each finite algebraic structure comes with a lattice of [[congruences|DCSXXT4SER]]: the ways of gluing its elements together that respect the operations.
The question is whether every finite lattice can be obtained in this way from some finite structure.
If one allows infinite structures, the answer is yes.
For finite structures nobody knows, and the question turns out to be the same as a question about subgroups of finite groups.