Roman Sikorski

Boolean Algebras (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 240
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783642858208
• Artikelnr.: 53155914

Inhaltsangabe

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.