Finite lattices and congruences: The good, the bad, and the ugly
by
George Gratzer
University of Manitoba
Coauthors: E. Thomas Schmidt

Finite lattices and congruences:
The good, the bad, and the ugly

G. Grätzer and E. T. Schmidt

Many of our earliest papers dealt with congruences and ideals of lattices. There were two major themes.

1. Lattice congruences are not as nice as congruences of groups and rings: There is no single class, in general, determining a congruence. We were interested when is there such a class and which types of ideals occur naturally in these discussions. This lead, in particular, to the introduction of the concept of standard ideals.

2. 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 the stronger form: We proved that the finite lattice L constructed was sectionally complemented.

We have been publishing papers on these two topics for 45 years. In this lecture, we are going to review some of our results. Many of them deal with the second topic, 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 much 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 congruence-preserving extensions.

We shall pay special attention to finite, sectionally complemented lattices. It is really odd that many of these constructions go through ``freely generated'' join-semilattices.

We shall conclude with some recent results on the spectrum of a congruence of a finite, sectionally complemented lattice, measuring the sizes of the congruence classes. With very few restrictions, this can be as bad as we wish.

Date received: January 29, 2002