Boolean Algebras

ISBN-13: 9783642858222 / Angielski / Miękka / 2012 / 240 str.

Roman Sikorski

Opis

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."

Spis treści

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.