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Partial Results & Related Problems

While the general FLRP remains open, significant progress has been made on special cases and related questions.

Known Representable Classes

Certain classes of finite lattices are known to be representable by finite algebras. These partial victories help delineate the boundary of the problem.

  • Finite Distributive Lattices: This was the first major result, proven by Dilworth.
  • Lattices with No Three-Element Antichains: A result from J. Snow (2000).

The Congruence Lattice Problem (CLP)

It's crucial to distinguish the FLRP from the related, but now solved, Congruence Lattice Problem (CLP).

  • CLP asked: Is every distributive algebraic lattice (possibly infinite) the congruence lattice of some lattice (possibly infinite)?
  • Status: Solved negatively by Friedrich Wehrung in 2007.

The negative solution to the CLP serves as an important precedent, showing that not all "reasonable" representation questions in lattice theory have a positive answer. However, because it deals with infinite structures and a restricted class of representing algebras (lattices only), it does not directly solve the FLRP.