Ebook: Lectures on Boolean Algebras
Author: Paul R. Halmos (auth.)
- Tags: Mathematical Logic and Foundations
- Series: Undergraduate Texts in Mathematics
- Year: 1974
- Publisher: Springer-Verlag New York
- Edition: 1
- Language: English
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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 . . .
Content:
Front Matter....Pages i-vii
Boolean Rings....Pages 1-3
Boolean algebras....Pages 3-9
Fields of sets....Pages 9-12
Regular open sets....Pages 12-17
Elementary relations....Pages 17-21
Order....Pages 21-25
Infinite operations....Pages 25-31
Subalgebras....Pages 31-35
Homomorphisms....Pages 35-40
Free algebras....Pages 40-47
Ideals and filters....Pages 47-51
The homomorphism theorem....Pages 52-55
Boolean ?-algebras....Pages 55-60
The countable chain condition....Pages 61-64
Measure algebras....Pages 64-69
Atoms....Pages 69-72
Boolean spaces....Pages 72-76
The representation theorem....Pages 77-80
Duality for ideals....Pages 81-84
Duality for homomorphisms....Pages 84-90
Completion....Pages 90-97
Boolean ?-spaces....Pages 97-99
The representation of ?-algebras....Pages 100-104
Boolean measure spaces....Pages 104-108
Incomplete algebras....Pages 109-115
Products of algebras....Pages 115-118
Sums of algebras....Pages 119-122
Isomorphisms of factors....Pages 122-126
Isomorphisms of countable factors....Pages 126-130
Retracts....Pages 130-136
Projective algebras....Pages 137-140
Injective algebras....Pages 140-143
Epilogue....Pages 144-144
Back Matter....Pages 145-147
Content:
Front Matter....Pages i-vii
Boolean Rings....Pages 1-3
Boolean algebras....Pages 3-9
Fields of sets....Pages 9-12
Regular open sets....Pages 12-17
Elementary relations....Pages 17-21
Order....Pages 21-25
Infinite operations....Pages 25-31
Subalgebras....Pages 31-35
Homomorphisms....Pages 35-40
Free algebras....Pages 40-47
Ideals and filters....Pages 47-51
The homomorphism theorem....Pages 52-55
Boolean ?-algebras....Pages 55-60
The countable chain condition....Pages 61-64
Measure algebras....Pages 64-69
Atoms....Pages 69-72
Boolean spaces....Pages 72-76
The representation theorem....Pages 77-80
Duality for ideals....Pages 81-84
Duality for homomorphisms....Pages 84-90
Completion....Pages 90-97
Boolean ?-spaces....Pages 97-99
The representation of ?-algebras....Pages 100-104
Boolean measure spaces....Pages 104-108
Incomplete algebras....Pages 109-115
Products of algebras....Pages 115-118
Sums of algebras....Pages 119-122
Isomorphisms of factors....Pages 122-126
Isomorphisms of countable factors....Pages 126-130
Retracts....Pages 130-136
Projective algebras....Pages 137-140
Injective algebras....Pages 140-143
Epilogue....Pages 144-144
Back Matter....Pages 145-147
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