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AuthorSikorski, Roman. author
TitleBoolean Algebras [electronic resource] / by Roman Sikorski
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1969
Edition Third Edition
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Descript X, 240 p. online resource


There are two aspects to the theory of Boolean algebras; the algebraic and the set-theoretical. A Boolean algebra can be considered as a special kind of algebraic ring, or as a generalization of the set-theoretical notion of a field of sets. Fundamental theorems in both of these directions are due to M. H. STONE, whose papers have opened a new era in the developยญ ment of this theory. This work treats the set-theoretical aspect, with little mention being made of the algebraic one. The book is composed of two chapters and an appendix. Chapter I is devoted to the study of Boolean algebras from the point of view of finite Boolean operations only; a greater part of its contents can be found in the books of BIRKHOFF [2J and HERMES [1]. Chapter II seems to be the first systematic study of Boolean algebras with infinite Boolean operations. To understand Chapters I and II it suffices only to know fundamental notions from general set theory and set-theoretical topology. No knowยญ ledge of lattice theory or of abstract algebra is presumed. Less familiar topological theorems are recalled, and only a few examples use more advanced topological means; but these may be omitted. All theorems in both chapters are given with full proofs


I. Finite joins and meets -- ยง 1. Definition of Boolean algebras -- ยง 2. Some consequences of the axioms -- ยง 3. Ideals and filters -- ยง 4. Subalgebras -- ยง 5. Homomorphisms, isomorphisms -- ยง 6. Maximal ideals and filters -- ยง 7. Reduced and perfect fields of sets -- ยง 8. A fundamental representation theorem -- ยง 9. Atoms -- ยง 10. Quotient algebras -- ยง11. Induced homomorphisms between fields of sets -- ยง 12. Theorems on extending to homomorphisms -- ยง 13. Independent subalgebras. Products -- ยง 14. Free Boolean algebras -- ยง 15. Induced homomorphisms between quotient algebras -- ยง 16. Direct unions -- ยง 17. Connection with algebraic rings -- II. Infinite joins and meets -- ยง 18. Definition -- ยง 19. Algebraic properties of infinite joins and meets. (m, n)-distributivity. -- ยง 20. m-complete Boolean algebras -- ยง 21. m-ideals and m-filters. Quotient algebras -- ยง 22. m-homomorphisms. The interpretation in Stone spaces -- ยง 23. m-subalgebras -- ยง 24. Representations by m-fields of sets -- ยง 25. Complete Boolean algebras -- ยง 26. The field of all subsets of a set -- ยง27. The field of all Borel subsets of a metric space -- ยง28. Representation of quotient algebras as fields of sets -- ยง 29. A fundamental representation theorem for Boolean ?-algebras. m-representability -- ยง 30. Weak m-distributivity -- ยง 31. Free Boolean m-algebras -- ยง 32. Homomorphisms induced by point mappings -- ยง 33. Theorems on extension of homomorphisms -- ยง 34. Theorems on extending to homomorphisms -- ยง 35. Completions and m-completions -- ยง 36. Extensions of Boolean algebras -- ยง 37. m-independent subalgebras. The field m-product -- ยง 38. Boolean (m, n)-products -- ยง 39. Relation to other algebras -- ยง 40. Applications to mathematical logic. Classical calculi -- ยง 41. Topology in Boolean algebras. Applications to non-classical logic -- ยง 42. Applications to measure theory -- ยง 43. Measurable functions and real homomorphisms -- ยง 44. Measurable functions. Reduction to continuous functions -- ยง 45. Applications to functional analysis -- ยง 46. Applications to foundations of the theory of probability -- ยง 47. Problems of effectivity -- List of symbols -- Author Index

Mathematics Mathematical logic Mathematics Mathematical Logic and Foundations Mathematical Logic and Formal Languages


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