The theory of lattice-ordered monoids

Abstract

The aim of this thesis is to provide a detailed description of chapter XIV of Garrett Birkhoff's "Lattice Theory" (3rd Edition). This chapter, entitled Lattice - ordered Monoids, introduces the concept of partially and lattice ordered algebraic systems, develops their theory, and studies their applications. No attempt has been made to cover all the material included in the chapter, but rather to develop and expand certain sections of it. In Chapter 1 the general theory of partially and lattice ordered algebraic systems is presented. The concepts of partially ordered groups and lattice - ordered groups, the subject of Chapter XIII of Birkhoff's "Lattice Theory", are introduced but not developed. While basically intended to serve as an introduction and to include material required for the following chapters, this chapter also provides greater detail for some of the examples and proofs of Birkhoff's Chapter XIV.