Directions in Lattice Theory

This paper is about three problems raised by Bjarni Jónsson and their influence on the development and direction of lattice theory. In lattice theory, like most fields, certain problems have played an important role in its development. R. P. Dilworth’s solutions of some of the most difficult problems of the thirties certainly raised the level of lattice theory. His result [17] that every lattice can be embedded into a uniquely complemented lattice surprised most of the lattice theorists of the time. The excellent article [1] by M. Adams details the history of and subsequent work on this problem. Some of the problems that have had the greatest influence the direction lattice theory has taken were proposed by Jónsson. Many of these problems, as well as many of the problems dating back to the thirties and before, have been solved in the last two decades. In this article, I have chosen three of Jónsson’s problems which have had a particularly strong influence on the development of lattice theory. We will also discuss some problems that are still unsolved.