<E-Book>
Boolean Algebras / by Roman Sikorski
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics ; 25)
Edition | 3rd ed. 1969. |
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Publisher | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer |
Year | 1969 |
Language | English |
Size | X, 240 p : online resource |
Authors | *Sikorski, Roman author SpringerLink (Online service) |
Subjects | LCSH:Mathematical logic LCSH:Machine theory FREE:Mathematical Logic and Foundations FREE:Formal Languages and Automata Theory |
Notes | 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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E-Book | Location | Media type | Volume | Call No. | Status | Reserve | Comments | ISBN | Printed | Restriction | Designated Book | Barcode No. |
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E-Book | オンライン | 電子ブック |
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Springer eBooks | 9783642858208 |
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電子リソース |
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EB00232084 |
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Material Type | E-Book |
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Classification | LCC:QA8.9-10.3 DC23:511.3 |
ID | 4000110362 |
ISBN | 9783642858208 |
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