Finite lattices and congruences.: A survey

In the early forties, R. P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L. We have been publishing papers on this topic for 45 years. In this survey paper, we are going to review some of our results and a host of related results by others: Making L "nice". If being "nice" is an algebraic property such as being semimodular or sectionally complemented, then we have tried in many instances to prove a stronger form of these results by verifying that every finite lattice has a congruence-preserving extension that is "nice". We shall discuss some of the techniques we use to construct "nice" lattices and congruence-preserving extensions. We shall describe some results on the spectrum of a congruence of a finite sectionally complemented lattice, measuring the sizes of the congruence classes. It turns out that with very few restrictions, these can be as bad as we wish. We shall also review some results on simultaneous representation of two distributive lattices. We conclude with the "magic wand" construction, which holds out the promise of obtaining results that go beyond what can be achieved with the older techniques.