Finite Lattice Representation Problem
There exists a finite lattice which is not the congruence lattice of a finite algebra.
A well-known result of universal algebra states: every algebraic lattice is isomorphic to the congruence lattice of an algebra. Thus there is essentially no restriction on the shape of a congruence lattice of a general algebra. It is natural to ask whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a finite algebra? This fundamental question, asked over 40 years ago, is among the most elusive problems of universal algebra.
This question is important because, until it is answered, we lack something very basic in our understanding of algebras -- namely, if we assume an algebra is finite, does this place any restriction on the shape of its congruence lattice? If so, then finite algebras are fundamentally different from infinite algebras in this sense.
[GS] Gratzer and Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34-59.
[P5] Palfy and Pudlak. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis, 11 (1980), 22–27.
[PT] Pudlak and Tuma, Every finite lattice can be embedded in the lattice of all equivalences over a finite set. Algebra Universalis, 10 (1980), 74–95.
* indicates original appearance(s) of problem.