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Author
Professor Leslie Cohn, Ph.D.
The Citadel
Department of Mathematics and Computer Science
171 Moultrie Street, Charleston, SC 29409, USA
url: http://155.225.10.78/lchp.htm, http://www.mathcs.citadel.edu
email: Cohnl@Citadel.edu
Abstract
This monograph induces a new class of quasi-orders, Gorenstein quasi-orders. Gorenstein quasi-orders are analogs of Gorenstein orders, which are a class of rings that generalize the rings of integers of algebraic number fields.
In the first chapter of this work, we give some very simple examples of Gorenstein quasi-orders. We then describe a functorial construction, the Gorenstein lift, which associates to a arbitrar quasi-order on a set X a Gorenstein quasi-order on the cartesian product X×Z of X with the set of integers Z. We call the simple examples together with Gorenstein lifts "special" Gorenstein quasi-orders. We then prove our main result on the structure/classification of Gorenstein quasi-orders, namely, that every Gorenstein quasi-order is a generalized sum over a total order of a family of special Gorenstein quasi-orders. Analysis of the isomorphisms of Gorenstein quasi-orders uncovers the existence of an important equivalence relation for quasi-orders, which we call Gorenstein equivalence, and whose properties we investigate in detail.
In the second chapter, we apply the machinery developed in Chapter 1 to obtain new generalized sum decompositions for arbitrary quasi-orders, where restrictions are placed on the summands.
The final chapter returns to the theory of Gorenstein quasi-orders. We describe another functorial construction of Gorenstein quasi-orders, which like the Gorenstein lift introduced in Chapter 1, associates to each quasi-order on a set X a Gorenstein quasi-order on X×Z. We call the new construction the "second Gorenstein lift." Chapter 3 makes explicit the decomposition of the second Gorenstein lift of a quasi-order as a generalized sum of special Gorenstein quasi-orders, whose existence follows from the structure theory of Gorenstein quasi-orders presented in Chapter 1.
AMS Classification: 06A06
Key words: coequalizer, coessential quasi-order, essential quasi-order, equalizer, generalized summation, Gorenstein quasi-order, Gorenstein lift, Gorenstein equivalence, induction of quasi-orders, invertible relation, second Gorenstein lift, strong coequalizer, strongly coessential quasi-order, weak equivalence
Contents
| Introduction | v | |
|---|---|---|
| Chapter 1. The Classification of Gorenstein Quasi-Orders | 1 | |
| 1.1. | Introduction | 1 |
| 1.2. | The Theory of Invertibility of Relations | 3 |
| 1.3. | Gorenstein Quasi-Orders | 38 |
| 1.4. | The Gorenstein Lift of a Quasi-Order | 43 |
| 1.5. | Generalized Summation of Quasi-Orders | 47 |
| 1.6. | Special Gorenstein Quasi-Orders | 55 |
| 1.7. | Cyclic Gorenstein Partial Orders | 57 |
| 1.8. | Cyclic Subsets of Gorenstein Partial Orders | 62 |
| 1.9. | The Fundamental Cone of a Set | 71 |
| 1.10. | Elementary Gorenstein Partial Orders | 78 |
| 1.11. | Structure of Gorenstein Partial Orders | 81 |
| 1.12. | Classification of Gorenstein Quasi-Orders | 87 |
| 1.13. | Isomorphisms of Gorenstein Quasi-Orders | 104 |
| 1.14. | The Automorphism Group of a Gorenstein Quasi-Order | 111 |
| 1.15. | Isomorphisms of Special Gorenstein Partial Orders | 122 |
| 1.16. | Gorenstein Equivalence of Quasi-Orders | 127 |
| 1.17. | The Gorenstein Equivalence Class of a Quasi-Order | 130 |
| Chapter 2. On the Representation of Quasi-Orders as Generalized Sums | 147 | |
| 2.1. | Introduction | 147 |
| 2.2. | Equivalence Relations on Q(X) | 151 |
| 2.3. | Generalized Summation of Quasi-Orders and Gorenstein Equivalence | 160 |
| 2.4. | Quasi-Orders and Equivalence Relations on X | 165 |
| 2.5. | Some Classes of Partial Orders | 203 |
| 2.6. | Statement of the Main Results | 219 |
| 2.7. | Proofs of the Main Lemmas | 227 |
| 2.8. | The Equalizer of an Induced Quasi-Order | 238 |
| 2.9. | The Strong Coequalizer of an Induced Quasi-Order | 242 |
| 2.10. | The Coequalizer of an Induced Quasi-Order | 246 |
| Chapter 3. The Second Gorenstein Lift of a Quasi-Order | 261 | |
| 3.1. | Introduction | 261 |
| 3.2. | Definition of the Quasi-order h(π). | |
| 3.3. | The Modified Tarski Generalized Sum Decomposition of a Quasi-Order | 269 |
| 3.4. | The Equivalence Relation E(π) | 278 |
| 3.5. | Study of h(π) when h^(π)= X × X | 283 |
| Appendix A. A Correspondence between Lattices and Relations | 307 | |
| A.1. | Split Relations | 307 |
| A.2. | Split Lattices in Matrix Algebras | 316 |
| Bibliography | 323 | |
| Index | 325 | |
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