Boolean algebras

Title page......Page 1
Series......Page 3
Copyright page......Page 4
Dedication......Page 5
Preface......Page 7
Preface to the 3rd edition......Page 8
Contents......Page 9
Terminology and notation......Page 11
§ 1. Definition of Boolean algebras......Page 13
§ 2. Some consequences of the axioms......Page 16
§ 3. Ideals and filters......Page 21
§ 4. Subalgebras......Page 23
§ 5. Homomorphisms, isomorphisms......Page 25
§ 6. Maximal ideals and filters......Page 27
§ 7. Reduced and perfect fields of sets......Page 30
§ 8. A fundamental representation theorem......Page 33
§ 9. Atoms......Page 37
§ 10. Quotient algebras......Page 39
§ 11. Induced homomorphisms between fields of sets......Page 42
§ 12. Theorems on extending to homomorphisms......Page 45
§ 13. Independent subalgebras. Products......Page 49
§ 14. Free Boolean algebras......Page 52
§ 15. Induced homomorphisms between quotient algebras......Page 55
§ 16. Direct unions......Page 60
§ 17. Connection with algebraic rings......Page 61
§ 18. Definition......Page 64
§ 19. Algebraic properties of infinite joins and meets, $(\\mathfrak{m}, \\mathfrak{n})$-distributivity......Page 69
§ 20. $\\mathfrak{m}$-complete Boolean algebras......Page 75
§ 21. $\\mathfrak{m}$-ideals and $\\mathfrak{m}$-filters. Quotient algebras......Page 84
§ 22. $\\mathfrak{m}$-homomorphisms. The interpretation in Stone spaces......Page 91
§ 23. $\\mathfrak{m}$-subalgebras......Page 101
§ 24. Representations by $\\mathfrak{m}$-fields of sets......Page 107
§ 25. Complete Boolean algebras......Page 115
§ 26. The field of all subsets of a set......Page 120
§ 27. The field of all Borel subsets of a metric space......Page 124
§ 28. Representation of quotient algebras as fields of sets......Page 125
§ 29. A fundamental representation theorem for Boolean $\\sigma$-algebras. $\\mathfrak{m}$-representability......Page 127
§ 30. Weak $\\mathfrak{m}$-distributivity......Page 137
§ 31. Free Boolean $\\mathfrak{m}$-algebras......Page 141
§ 32. Homomorphisms induced by point mappings......Page 146
§ 33. Theorems on extension of homomorphisms......Page 151
§ 34. Theorems on extending to homomorphisms......Page 154
§ 35. Completions and $\\mathfrak{m}$-completions......Page 162
§ 36. Extensions of Boolean algebras......Page 175
§ 37. $\\mathfrak{m}$-independent subalgebras. The field $\\mathfrak{m}$-product......Page 182
§ 38. Boolean $(\\mathfrak{m}, \\mathfrak{n})$-products......Page 185
§ 39. Relation to other algebras......Page 201
§ 40. Applications to mathematical logic. Classical calculi......Page 204
§ 41. Topology in Boolean algebras. Applications to non-classical logic......Page 208
§ 42. Applications to measure theory......Page 211
§ 43. Measurable functions and real homomorphisms......Page 214
§ 44. Measurable functions. Reduction to continuous functions......Page 216
§ 45. Applications to functional analysis......Page 217
§ 46. Applications to foundations of the theory of probability......Page 218
§ 47. Problems of effectivity......Page 220
Bibliography......Page 222
List of symbols......Page 241
Author Index......Page 242
Subject Index......Page 245