Heyting Algebras - Duality Theory
von: Leo Esakia, Guram Bezhanishvili, Wesley H. Holliday
Springer-Verlag, 2019
ISBN: 9783030120962
Sprache: Englisch
107 Seiten, Download: 1683 KB
Format: PDF, auch als Online-Lesen
| Foreword | 7 | ||
| Contents | 9 | ||
| Editors’ Note | 10 | ||
| Introduction | 12 | ||
| 1 Preliminary Notions and Necessary Facts | 15 | ||
| 1.1 Universal Algebra | 15 | ||
| 1.2 Categories | 16 | ||
| 1.3 Topologies | 18 | ||
| 1.4 Ordered Sets and Clusters | 20 | ||
| 1.5 Heyting Lattices | 25 | ||
| References | 28 | ||
| 2 Heyting Algebras and Closure Algebras | 29 | ||
| 2.1 Heyting Algebras | 29 | ||
| 2.2 Closure Algebras | 31 | ||
| 2.3 Modal Systems and Superintuitionistic Logics | 35 | ||
| 2.4 Filters and Congruences | 38 | ||
| 2.5 Skeletal Closure Algebras | 42 | ||
| References | 54 | ||
| 3 Duality Theory: Hybrids | 1 | ||
| 3.1 The Hybrid of Topology (Stone) and Order (Kripke) | 55 | ||
| 3.2 Fundamental Properties of Hybrids | 61 | ||
| 3.3 The Category of Hybrids and Hybrid Maps | 68 | ||
| 3.4 The Category of Heyting Algebras and the Category of Strict Hybrids | 75 | ||
| 3.5 Grzegorczyk Algebras | 82 | ||
| References | 89 | ||
| Appendix | 90 | ||
| A.1 On Subdirectly Irreducible Heyting Algebras and Subdirect Products | 90 | ||
| A.2 Hybrid Formulation of Glivenko's Theorem | 91 | ||
| A.3 Weak Decompositions of Boolean Algebras and Heyting Algebras | 92 | ||
| A.4 The Lattice of Varieties of Heyting Algebras | 94 | ||
| A.5 Representable Ordered Sets: Grätzer's Problems | 94 | ||
| A.6 MacNeille Completions | 95 | ||
| A.7 The Category of Heyting Algebras is Balanced | 95 | ||
| A.8 Equationally Non-conservative Operations | 96 | ||
| A.9 Boolean Cascades | 98 | ||
| A.10 Cantor's Scattered Spaces | 100 | ||
| A.11 Symmetric Heyting algebras | 101 | ||
| References | 102 | ||
| Index | 103 |