Lectures on Boolean Algebras

by Steven Givant, P.R. Halmos

Overview

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 . . .

Product Details

ISBN-13:	9780387900940
Publisher:	Springer New York
Publication date:	01/01/1974
Series:	Undergraduate Texts in Mathematics
Edition description:	Softcover reprint of the original 1st ed. 1974
Pages:	148
Product dimensions:	5.98(w) x 9.02(h) x 0.01(d)

About the Author

Paul R. Halmos (1916–2006) was a prominent American mathematician who taught at the University of Chicago, the University of Michigan, and other schools and made significant contributions to several areas of mathematics, including mathematical logic, ergodic theory, functional analysis, and probability theory. His other Dover books are Algebraic Logic, Finite-Dimensional Vector Spaces, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Lectures on Ergodic Theory, and Naive Set Theory.

Table of Contents

1 Boolean rings.- 2 Boolean algebras.- 3 Fields of sets.- 4 Regular open sets.- 5 Elementary relations.- 6 Order.- 7 Infinite operations.- 8 Subalgebras.- 9 Homomorphisms.- 10 Free algebras.- 11 Ideals and filters.- 12 The homomorphism theorem.- 13 Boolean—-algebras.- 14 The countable chain condition.- 15 Measure algebras.- 16 Atoms.- 17 Boolean spaces.- 18 The representation theorem.- 19 Duality for ideals.- 20 Duality for homomorphisms.- 21 Completion.- 22 Boolean—-spaces.- 23 The representation of σ-algebras.- 24 Boolean measure spaces.- 25 Incomplete algebras.- 26 Products of algebras.- 27 Sums of algebras.- 28 Isomorphisms of factors.- 29 Isomorphisms of countable factors.- 30 Retracts.- 31 Projective algebras.- 32 Injective algebras.- Epilogue.