Lectures on Boolean Algebras by Paul R. Halmos

By Paul R. Halmos

IN 1959 I lectured on Boolean algebras on the college of Chicago. A mimeographed model of the notes on which the lectures have been dependent circulated for roughly years; this quantity includes these notes, corrected and revised. many of the corrections have been advised via Peter Crawley. to pass judgement on through his unique and specified feedback, he should have learn each observe, checked each reference, and weighed each argument, and i'm lIery thankful to hirn for his support. this isn't to assert that he's to be held accountable for the imperfec­ tions that stay, and, specifically, I on my own am liable for all expressions of non-public opinion and irreverent view­ aspect. P. R. H. Ann Arbor, Michigan ] anuary, 1963 Contents part web page 1 1 Boolean jewelry ............................ . 2 Boolean algebras ......................... . three nine three Fields of units ............................ . four standard open units . . . . . . . . . . . . . . . . . . . 12 . . . . . . five common family members. . . . . . . . . . . . . . . . . . 17 . . . . . 6 Order. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . 7 endless operations. . .. . . . . . . . . . . . . . . . . 25 . . . . . eight Subalgebras . . . . . . . . . . . . . . . . . . . . .. . . . 31 . . . . . . nine Homomorphisms . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . 10 unfastened algebras . . . . . . . . . . . . . . . . . . . . . . forty . . . . . . . eleven beliefs and filters. . . . . . . . . . . . . . . . . . . . forty seven . . . . . . 12 The homomorphism theorem. . . . . . . . . . . . .. . . fifty two . . thirteen Boolean a-algebras . . . . . . . . . . . . . . . . . . fifty five . . . . . . 14 The countable chain situation . . . . . . . . . . . . sixty one . . . 15 degree algebras . . . . . . . . . . . . . . . . . . . sixty four . . . . . . . sixteen Atoms.. . . . .. . . . . .. .. . . . ... . . . . .. . . ... . . .. sixty nine 17 Boolean areas . . . . . . . . . . . . . . . . . . . . seventy two . . . . . . . 18 The illustration theorem. . . . . . . . . . . . . . seventy seven . . . 19 Duali ty for beliefs . . . . . . . . . . . . . . . . . .. . . eighty one . . . . . 20 Duality for homomorphisms . . . . . . . . . . . . . . eighty four . . . . 21 crowning glory . . . . . . . . . . . . . . . . . . . . . . . ninety . . . . . . . . 22 Boolean a-spaces . . . . . . . . . . . . . . . . . .. . . ninety seven . . . . . 23 The illustration of a-algebras . . . . . . . . .. . . a hundred . 24 Boolean degree areas . . . . . . . . . . . . . .. . . 104 . . . 25 Incomplete algebras . . . . . . . . . . . . . . . .. . . 109 . . . . . 26 items of algebras . . . . . . . . . . . . . . . .. . . a hundred and fifteen . . . . 27 Sums of algebras . . . . . . . . . . . . . . . . . .. . . 119 . . . . . 28 Isomorphisms of things . . . . . . . . . . . . . .. . . 122 . . .

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An ideal generated by a singleton Ip 1 is called a princ ipal ideal; it consists of all the subelements of p. The concepts of subalgebra and homomorphism are in a certain obvious sense self-dual; the concept of ideal is not. The dual concept is defined as follows. A Boolean filter in a Boolean algebra B is a subset N of B such that (4) 1 E N, (5) if p E N and q E N, then p q E N, (6) if p E N and q E B, then p V q E N• 1\ The condition (4) can be replaced by the condition that N be not empty. The condition (6) can be replaced by if p E N and p ;:;; q, then q E N• Ideals and Filters 51 Neither of these replacements will alter the concept being defined.

Since P(i, +1) V P(i, -1) is equal to the entire space (0, 1) for each i, it follows that the left side of (1) is the unit element of the algebra under consideration. A moment's reflection on the binary expansions of real numbers shows that 30 Lectures on Boolean Algebras ni P(i, a(i)) consists of at most one point, whatever the function a in ] I may be; i t follows that the right s ide of (1) is the zero element of our algebra. Exercises (1) 1s a complete field of subsets of a set X the same as the field of all subsets of X?

The proof is elementary. Ideals and Filters 49 Here is a general and useful remark about homomorphisms and their kerneis: a necessary and sufficient condition that a homomorphism be a monomorphism (one-to-one) is that its kernel be ! 0 l. Proof of necessity: if [is one-to-one and f(p) = 0, then [(p) = f(0), and, therefore, ,p = O. Proof of sufficiency: if the kernel of [is 101 and if [(p) = f(q), then f(p + q) [(p) + [(q) = 0, so that p + q = 0, and this me ans that p = q. Every example of a homomorphism (such as the ones we saw in §9) gives rise to an example of an ideal, namely its kernel.

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