Introduction to Boolean Algebras

589 Pages·2009·5.03 MB·English

by  Paul Halmos

Undergraduate Texts in Mathematics
Editors:
S.Axler
K.A.Ribet
Undergraduate Texts in Mathematics
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Steven Givant (cid:129) Paul Halmos
Introduction to Boolean
Algebras
ABC
StevenGivant PaulHalmos
MillsCollege (Deceased)
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andComputerScience
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S.Axler K.A.Ribet
MathematicsDepartment DepartmentofMathematics
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ISSN:0172-6056
ISBN:978-0-387-40293-2 e-ISBN:978-0-387-68436-9
DOI:10.1007/978-0-387-68436-9
LibraryofCongressControlNumber: 2008939810
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Contents
Preface ix
1 Boolean Rings 1
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Boolean Algebras 8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Boolean Algebras Versus Rings 14
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 The Principle of Duality 20
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Fields of Sets 24
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Elementary Relations 31
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 Order 38
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8 Infinite Operations 45
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9 Topology 53
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
v
vi Introduction to Boolean Algebras
10 Regular Open Sets 66
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
11 Subalgebras 74
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
12 Homomorphisms 89
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
13 Extensions of Homomorphisms 105
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
14 Atoms 117
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
15 Finite Boolean Algebras 127
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
16 Atomless Boolean Algebras 134
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
17 Congruences and Quotients 142
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
18 Ideals and Filters 149
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
19 Lattices of Ideals 164
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
20 Maximal Ideals 171
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
21 Homomorphism and Isomorphism Theorems 178
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
22 The Representation Theorem 188
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
23 Canonical Extensions 193
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Contents vii
24 Complete Homomorphisms and Complete Ideals 200
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
25 Completions 214
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
26 Products of Algebras 221
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
27 Isomorphisms of Factors 243
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
28 Free Algebras 256
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
29 Boolean σ-algebras 268
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
30 The Countable Chain Condition 282
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
31 Measure Algebras 288
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
32 Boolean Spaces 300
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
33 Continuous Functions 312
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
34 Boolean Algebras and Boolean Spaces 326
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
35 Duality for Ideals 338
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
36 Duality for Homomorphisms 347
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
37 Duality for Subalgebras 359
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
viii Introduction to Boolean Algebras
38 Duality for Completeness 368
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
39 Boolean σ-spaces 373
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
40 The Representation of σ-algebras 378
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
41 Boolean Measure Spaces 384
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
42 Incomplete Algebras 390
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
43 Duality for Products 396
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
44 Sums of Algebras 422
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
45 Isomorphisms of Countable Factors 439
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Epilogue 447
A Set Theory 448
B Hints to Selected Exercises 474
References 536
Index 542
Preface
The theory of Boolean algebras was created in 1847 by the English mathe-
matician George Boole. He conceived it as a calculus (or arithmetic) suitable
for a mathematical analysis of logic. The form of his calculus was rather
different from the modern version, which came into being during the pe-
riod 1864–1895 through the contributions of William Stanley Jevons, Augus-
tus De Morgan, Charles Sanders Peirce, and Ernst Schro¨der. A foundation
of the calculus as an abstract algebraic discipline, axiomatized by a set of
equations, and admitting many different interpretations, was carried out by
Edward Huntington in 1904.
Only with the work of Marshall Stone and Alfred Tarski in the 1930s,
however, did Boolean algebra free itself completely from the bonds of logic
and become a modern mathematical discipline, with deep theorems and im-
portantconnectionstoseveralotherbranchesofmathematics,includingalge-
bra,analysis,logic,measuretheory,probabilityandstatistics,settheory,and
topology. For instance, in logic, beyond its close connection to propositional
logic, Boolean algebra has found applications in such diverse areas as the
proof of the completeness theorem for first-order logic, the proof of the L(cid:3) o´s
conjecture for countable first-order theories categorical in power, and proofs
of the independence of the axiom of choice and the continuum hypothesis
in set theory. In analysis, Stone’s discoveries of the Stone–Cˇech compacti-
fication and the Stone–Weierstrass approximation theorem were intimately
connected to his study of Boolean algebras. Countably complete Boolean
algebras (also called σ-algebras) and countably complete fields of sets (also
called σ-fields) play a key role in the foundations of measure theory. Out-
side the realm of mathematics, Boolean algebra has found applications in
such diverse areas as anthropology, biology, chemistry, ecology, economics,
sociology, and especially computer science and philosophy. For example, in
computer science, Boolean algebra is used in electronic circuit design (gating
networks), programming languages, databases, and complexity theory.
ix