Description

To the Instructor I want to begin by giving credit where credit is due — the exercises presented in these notes got their start with Constantine Tsinakis at Vanderbilt University. I had the good fortune of writing my dissertation under Professor Tsinakis in the late 1980’s, and it was in this setting that I benefitted from his carefully chosen problems. Over the decades that followed, I added many additional problems to suit the particular needs of my students but never improved on the core exercises I first explored under Constantine’s guidance. Readers who also studied under Professor Tsinakis will no doubt see his signature written subtly across these notes, especially in the material on distributive lattices. I am also indebted to Zack French — a particularly precocious undergraduate — who not only worked through these notes in a guided independent study, but also edited the collection, corrected errors, and provided a number of new exercises. His comments to potential students appear below. This collection develops basic order theory through a series of problems that are designed to move the reader through the concepts. A few proofs are scattered throughout to provide the reader with some clues regarding the techniques and style of order-theoretic arguments. The exercises presented in these notes have been used as the foundation study for four Masters’ Theses, two graduate courses on order theory, and a number of independent studies. These exercises may be used in a classroom setting, or as a supervised independent study. However they are used, it is critical that the socratic method play a significant role. Indeed, these exercises were designed with individual reflection and struggle, followed by discussion, debate, and critique as the intended vehicle for moving the reader forward through the study. While it is certainly helpful for the study leader / instructor to have some knowledge of order theory, this is not required. It is important howiii Foreword iv ever, that the study leader / instructor be well-versed in the art of formal proof. If these notes are used in a guided independent study, the instructor should plan on meeting with the student once a week. The student should present her work to the instructor; and the two should discuss the work together and progress toward an acceptable end product. As an independent study, I have found that it will take three to four months to complete all of the exercises. If these notes are used in a classroom setting, then some variation of the “Moore method” should be used as the pedagogical model. While it is certainly possible to use a “pure” Moore method model and simply “turn students loose” on the exercises and let them compete, I have found it best to assign students exercises in advance and have them present their work to the class. This approach assures that steady progress will be made and encourages lively discussion as students struggle to understand work presented on a problem that they may not have investigated. If undergraduates or beginning graduate students are part of the class, I have also found it helpful to give weekly definitions quizzes to keep the class abreast of the ever-growing list of technical terms and concepts. In a class of experienced graduate students, formal exams are usually not necessary as the reflectionpresentation-discussion cycle is sufficient for evaluation of student progress. In the classes where I felt the need to use testing, I have found that one exam at mid-term and one at the end of the semester, coupled with the weekly quizzes and daily presentations, have always provided me with ample data for summative purposes.