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＜電子ブック＞
Boolean Algebras / by Roman Sikorski
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics ; 25)

版	3rd ed. 1969.
出版者	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer
出版年	1969
本文言語	英語
大きさ	X, 240 p : online resource
著者標目	*Sikorski, Roman author SpringerLink (Online service)
件　名	LCSH:Mathematical logic LCSH:Machine theory FREE:Mathematical Logic and Foundations FREE:Formal Languages and Automata Theory
一般注記	I. Finite joins and meets -- § 1. Definition of Boolean algebras -- § 2. Some consequences of the axioms -- § 3. Ideals and filters -- § 4. Subalgebras -- § 5. Homomorphisms, isomorphisms -- § 6. Maximal ideals and filters -- § 7. Reduced and perfect fields of sets -- § 8. A fundamental representation theorem -- § 9. Atoms -- § 10. Quotient algebras -- §11. Induced homomorphisms between fields of sets -- § 12. Theorems on extending to homomorphisms -- § 13. Independent subalgebras. Products -- § 14. Free Boolean algebras -- § 15. Induced homomorphisms between quotient algebras -- § 16. Direct unions -- § 17. Connection with algebraic rings -- II. Infinite joins and meets -- § 18. Definition -- § 19. Algebraic properties of infinite joins and meets. (m, n)-distributivity. -- § 20. m-complete Boolean algebras -- § 21. m-ideals and m-filters. Quotient algebras -- § 22. m-homomorphisms. The interpretation in Stone spaces -- § 23. m-subalgebras -- § 24. Representations by m-fields of sets -- § 25. Complete Boolean algebras -- § 26. The field of all subsets of a set -- §27. The field of all Borel subsets of a metric space -- §28. Representation of quotient algebras as fields of sets -- § 29. A fundamental representation theorem for Boolean ?-algebras. m-representability -- § 30. Weak m-distributivity -- § 31. Free Boolean m-algebras -- § 32. Homomorphisms induced by point mappings -- § 33. Theorems on extension of homomorphisms -- § 34. Theorems on extending to homomorphisms -- § 35. Completions and m-completions -- § 36. Extensions of Boolean algebras -- § 37. m-independent subalgebras. The field m-product -- § 38. Boolean (m, n)-products -- § 39. Relation to other algebras -- § 40. Applications to mathematical logic. Classical calculi -- § 41. Topology in Boolean algebras.Applications to non-classical logic -- § 42. Applications to measure theory -- § 43. Measurable functions and real homomorphisms -- § 44. Measurable functions. Reduction to continuous functions -- § 45. Applications to functional analysis -- § 46. Applications to foundations of the theory of probability -- § 47. Problems of effectivity -- List of symbols -- Author Index 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 develop­ 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 know­ 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 HTTP:URL=https://doi.org/10.1007/978-3-642-85820-8

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