Order on abelian groups - Leo, Chee Wee (Nanyang Technological University, 2023)
This thesis explores the fundamental concepts of Group Theory and Order Theory, focusing on their application in the study of Ordered Monoids and Abelian Groups. Chapter 1 serves as an introduction, providing an overview of the paper’s scope and objectives. Chapter 2 delves into the historical and background context, outlining the evolution and significance of the theories under consideration. Chapter 3 offers a comprehensive examination of Group Theory, analyzing its principles and properties in detail. The discussion encompasses key topics such as group operations, subgroups, among others. Moving on, chapter 4 discusses abelian groups. In Chapter 5, the thesis delves into Order Theory, exploring the foundations of partially ordered sets, Hasse, and their associated properties. Important definitions such as greatest lower bound and least upper bound will be discussed as well. Chapter 6 represents a focal point of the research, delving into the intricate workings of Ordered Monoids. The chapter investigates the structural characteristics of these mathematical systems, analyzing their elements, operations, and partial order relations. H ̈older’s Theorem is introduced as well. Lastly, the thesis ends with a discussion on Levi’s theorem.