Lectures on Boolean Algebras

1. Front Matter

   Pages i-vii

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2. Boolean Rings

   • Paul R. Halmos

   Pages 1-3

3. Boolean algebras

   • Paul R. Halmos

   Pages 3-9

4. Fields of sets

   • Paul R. Halmos

   Pages 9-12

5. Regular open sets

   • Paul R. Halmos

   Pages 12-17

6. Elementary relations

   • Paul R. Halmos

   Pages 17-21

7. Order

   • Paul R. Halmos

   Pages 21-25

8. Infinite operations

   • Paul R. Halmos

   Pages 25-31

9. Subalgebras

   • Paul R. Halmos

   Pages 31-35

10. Homomorphisms

    • Paul R. Halmos

    Pages 35-40

11. Free algebras

    • Paul R. Halmos

    Pages 40-47

12. Ideals and filters

    • Paul R. Halmos

    Pages 47-51

13. The homomorphism theorem

    • Paul R. Halmos

    Pages 52-55

14. Boolean σ-algebras

    • Paul R. Halmos

    Pages 55-60

15. The countable chain condition

    • Paul R. Halmos

    Pages 61-64

16. Measure algebras

    • Paul R. Halmos

    Pages 64-69

17. Atoms

    • Paul R. Halmos

    Pages 69-72

18. Boolean spaces

    • Paul R. Halmos

    Pages 72-76

19. The representation theorem

    • Paul R. Halmos

    Pages 77-80

20. Duality for ideals

    • Paul R. Halmos

    Pages 81-84

IN 1959 I lectured on Boolean algebras at the University of Chicago. A mimeographed version of the notes on which the lectures were based circulated for about two years; this volume contains those notes, corrected and revised. Most of the corrections were suggested by Peter Crawley. To judge by his detailed and precise suggestions, he must have read every word, checked every reference, and weighed every argument, and I am lIery grateful to hirn for his help. This is not to say that he is to be held responsible for the imperfec­ tions that remain, and, in particular, I alone am responsible for all expressions of personal opinion and irreverent view­ point. P. R. H. Ann Arbor, Michigan ] anuary, 1963 Contents Section Page 1 1 Boolean rings ............................ . 2 Boolean algebras ......................... . 3 9 3 Fields of sets ............................ . 4 Regular open sets . . . . . . . . . . . . . . . . . . . 12 . . . . . . 5 Elementary relations. . . . . . . . . . . . . . . . . . 17 . . . . . 6 Order. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . 7 Infinite operations. . .. . . . . . . . . . . . . . . . . 25 . . . . . 8 Subalgebras . . . . . . . . . . . . . . . . . . . . .. . . . 31 . . . . . . 9 Homomorphisms . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . 10 Free algebras . . . . . . . . . . . . . . . . . . . . . . 40 . . . . . . . 11 Ideals and filters. . . . . . . . . . . . . . . . . . . . 47 . . . . . . 12 The homomorphism theorem. . . . . . . . . . . . .. . . 52 . . 13 Boolean a-algebras . . . . . . . . . . . . . . . . . . 55 . . . . . . 14 The countable chain condition . . . . . . . . . . . . 61 . . . 15 Measure algebras . . . . . . . . . . . . . . . . . . . 64 . . . . . . . 16 Atoms.. . . . .. . . . . .. .. . . . ... . . . . .. . . ... . . .. 69 17 Boolean spaces . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . 18 The representation theorem. . . . . . . . . . . . . . 77 . . . 19 Duali ty for ideals . . . . . . . . . . . . . . . . . .. . . 81 . . . . . 20 Duality for homomorphisms . . . . . . . . . . . . . . 84 . . . . 21 Completion . . . . . . . . . . . . . . . . . . . . . . . 90 . . . . . . . . 22 Boolean a-spaces . . . . . . . . . . . . . . . . . .. . . 97 . . . . . 23 The representation of a-algebras . . . . . . . . .. . . 100 . 24 Boolean measure spaces . . . . . . . . . . . . . .. . . 104 . . . 25 Incomplete algebras . . . . . . . . . . . . . . . .. . . 109 . . . . . 26 Products of algebras . . . . . . . . . . . . . . . .. . . 115 . . . . 27 Sums of algebras . . . . . . . . . . . . . . . . . .. . . 119 . . . . . 28 Isomorphisms of factors . . . . . . . . . . . . . .. . . 122 . . .

• Boolean algebra
• Boolesche Algebra
• Factor
• Finite
• Morphism
• Volume
• algebra
• boundary element method
• duality
• homomorphism
• measure
• presentation
• sets
• theorem
• university

• Department of Mathematics, Indiana University, Bloomington, USA

  Paul R. Halmos