By J. Donald Monk

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"This ebook is an fundamental device for an individual operating in Boolean algebra, and can be steered for set-theoretic topologists." **- Zentralblatt MATH**

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**Example text**

Now x · -a =f. 0, so we can choose a non-zero b E A such that b :::; x · -a. Then a < a+ b:::; x, as desired. Similarly, A I -x is non-principal. 25. Suppose that A(x) is a proper minimal extension of A, and A I x is a principal ideal generated by an element a*. Set y = -a* · x. Then: (i} y tJ. A, and hence A(x) = A(y); (ii} for all a E A, y:::; a iff -a E Smp~; (iii} y is an atom of A(x); (iv} if D is dense in A, then (D U {y}) is dense in A(x). Proof. (i): Clearly y tJ. A, since otherwise y:::; a*, y = 0, x:::; a*, and x =a* EA.

First assume that y:::; a. Thus -a:::; a*+-x. Now -a= -a·a*+-a·-a*, and -a · -a* :::; -x, proving that -a E Smp~. 25 35 Second, assume that -a E Smp~. Say -a = b + c with b E A I x and c E A I -x. Then -a* · x · -a = -a* · b = 0, showing that y ::; a. (iii). Clearly y -=f. 0, by (i). Suppose that z::; y; we show that z = 0 or z = y. Say z = a·x+b· -x with a, bE A. Since y::; x, we have b· -x = 0. 20, either a E Smp~ or -a E Smp~. If -a E Smp~, then y::; a by (ii), hence y = z, as desired. Now suppose that a E Smp~.

5. A satisfy condition (o:). Suppose that (aa : o: < ~;;;-) is ::;*- increasing, I is an infinite subset ofw, and A ~f {aa: o: <~;;;-}is ::;*-unbounded on I. Then 9"1 A satisfies the ~;;;- -cc. Proof. Let (Pa : o: < ~;;;-) be a sequence of distinct elements of &J1A. We want to find distinct o:, f3 < ~~ such that Pa and Pf3 are compatible. Without loss of generality (pa : o: < ~;;;-) is a ~-system, say with kernel q. 6 49 Free products a, (3 with Po: \q and P/3 \q compatible; so wlog the Po: 's are pairwise disjoint.