Fundamental Notions of Lattice Theory

Abstract

This first chapter deals with those parts of lattice theory which are used later. It essentially contains only elementary results. A reader with some basic knowledge of lattice theory can go directly to Chapter 2. Then he can look up those parts of Chapter 1 with which he might feel not familiar enough, whenever references are stated. For this book, the most important example is the lattice of subspaces of a projective geometry. The verification that it has the various properties introduced and discussed in this chapter has to be postponed until projective geometries will be available in Chapter 2. In order to understand this chapter, some knowledge of posets, i.e. partially ordered sets, is necessary. In particular, the following notions are supposed to be known: partial and total order on a given set, upper and lower bounds, greatest and smallest elements, maximal and minimal elements. Finally, Zorn’s Lemma will be formulated, but not proved.