History and Key Milestones

The FLRP is built upon a rich history of research. Here are the key developments that led to the problem and have shaped its study.

Year	Key Figure(s)	Contribution
c. 1940	R. P. Dilworth	Proved every finite distributive lattice is the congruence lattice of a finite lattice.
1948	G. Birkhoff & O. Frink	Established that \(\text{Con } \mathbf{A}\) of any algebra \(\mathbf{A}\) is an algebraic lattice.
1963	G. Grätzer & E. T. Schmidt	Proved that every algebraic lattice is \(\text{Con } \mathbf{A}\) for some (possibly infinite) algebra. This set the stage for the FLRP.
1980	P. Pudlák & J. Tůma	Showed every finite lattice can be embedded into the lattice of all equivalence relations on a finite set.
1980	P. P. Pálfy & P. Pudlák	The Pálfy-Pudlák Theorem: Proved the FLRP is equivalent to a problem in finite group theory.
1980s-Present	DeMeo, Freese, Hobby, Jipsen, McKenzie, Snow, et al.	Development of Tame Congruence Theory; computational algebra approaches; focused search for counterexamples.
2007	F. Wehrung	Solved the related (but infinite) Congruence Lattice Problem (CLP) negatively.

The Pálfy-Pudlák Theorem: A Game-Changing Equivalence

The 1980 theorem by Pálfy and Pudlák was a watershed moment. It transformed the problem by providing a powerful, if equally difficult, alternative formulation:

A finite lattice \(L\) is representable as the congruence lattice of a finite algebra if and only if \(L\) is isomorphic to an interval in the subgroup lattice of a finite group.

This connected the abstract problem of congruence lattices to the concrete world of finite group theory, opening up entirely new avenues of attack.