Algorithms for Finite, Finitely Presented and Free Lattices

In this talk we will present and analyze the efficiency of various algorithms in lattice theory. For finite lattices this will include recognition of various properties such as subdirect irreducibility, semidistributivity, and boundedness (in the sense of McKenzie) as well as efficient algorithms for computing the congruence lattice. For free and finitely presented lattices we will discuss algorithms for such things as finding all the lower and upper covers of an element and recognizing if a finitely presented lattice is finite. Several interesting open problems will be given. We will also discuss how (computer implementations of) these algorithms have been used to prove results in various branches of lattice theory. I first became interested in computer programs to help with lattice theory when Nation and I were studying free lattices. Putting a lattice term into (Whitman) canonical form can be tedious, so we developed a program to do this and calculations in free lattices in general. This program was expanded to include finite and finitely presented lattices, congruence lattices, and automatic lattice drawing and proved extremely useful. As part of our book [12] on free lattices we decided to write a chapter on lattice algorithms for free lattices and those aspects of finite lattices related to free lattices. We expanded the chapter to include a general introduction to algorithms for lattices. We found the literature on this subject spread over both computer science and mathematical journals. Algorithms on directed graphs have been extensively developed and many are basic to ordered sets and algorithms from the theory of data bases play a role.