A primer on lattice theory and distributive lattices

Abstract

The notion of an order is one of great importance. Almost all facets of life utilise order structure, Mathematics especially so. In fact, many areas in Mathematics (and adjacent fields of study), depend on order explicitly or implicitly. In this dissertation, we will solidify our intuition about order by providing a formal framework which captures this intuition. Then we will focus on exploring, expanding and testing the limits of said framework. This will be done in such a way as to ensure the dissertation remains introductory in nature, whilst still providing the necessary detail for airtight arguments and pointers for further study. The dissertation will be split into four chapters. In the first chapter, the foundations of lattice theory will be built. These foundations will be accompanied by examples when necessary. In the second chapter, we will get close to Category Theory by exploring Free Lattices without entering into the realm of Category Theory. The third chapter, pivots the reader onto Distributive Lattices. In particular, a result linking: diagrams, free lattices and distributivity will allow us to characterize all distributive lattices. Moreover, a famous representation theorem of distributive lattice, which is due to G. Birkhoff and M. H. Stone, will be proven. Finally, the fourth chapter will provide the reader with further areas of exploration within lattice theory and one interesting application.