Heyting Algebras (eBook)

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