Height and Width of Superatomic Boolean Algebras

Cantor-Bendixson height and width of superatomic Boolean algebras is investigated and it is shown that (1) you don't need a Canadian tree to construct an u>,-thin-thick superatomic Boolean algebra; (2) c can be very large and for all k < c and all uncountable X, k < c, there are no K-thin-very tall, X-thin-tall, ic-very thin-thick, or X-thin-thick superatomic Boolean algebras on k. 0. Introduction. A superatomic Boolean algebra (henceforth abbreviated sBa) is a Boolean algebra in which every subalgebra (equivalently, every quotient algebra) is atomic. The class of sBa's is exactly the class of Boolean algebras whose Stone spaces are compact scattered; hence, the results of this paper have direct translations into the theory of compact scattered spaces. Let A' be a Boolean algebra. For each ordinal a we define the ath Cantor-Bendixson ideal Ja on X as follows: /0 ¥= 0 ; given Ja, let Ata(X) = {x: x/Ja is an atom of X/Ja) and let Ja + X be the ideal generated by Ja U Ata(X); given Jß for all ß < a, a a limit, let Ja = U/8<a//3. X is an sBa iff, for some a, X = Ja. Suppose X is an sBa. For each x g X we define rank(x) to be the least a with x g Ja + X Ja. The Cantor-Bendixson height of X, ht(.Y), is the least a such that X = Ja; note that ht(A') is always a successor ordinal. For each ordinal a, let wda(Ar) = \Ja + x/Ja\ (i.e. the number of atoms in X/Ja), and define the CantorBendixson width of X, wd^), to be the supremum of all wda( X). An sBa X is (a) K-thin iff wd( X) = k. (Note: thin = w-thin.) (b) K-thin-thick iff wda(A') = k for a < k and wdK(X) = k+. (Note: thin-thick = to,-thin-thick + Just's thin-thick.) (c) K-very-thin-thick iff wa(X) < k for a < k and wdK(X) = k+. (Note: very-thinthick = to,-very-thin-thick = Just's thin-thick.) (d) K-thin-very-thick iff wa(X) = k for a < k and wdK(X) = k . (e) K-thin-tall iff Jf is K-thin and ht( X) = k+. (Note: thin-tall = «-thin tall.) (f) K-thin-very tall iff Xis K-thin and ht(A') = k + +. (Note: thin very tall = «-thinvery tall.) The existence of thin-tall sBa's was shown by Rajagapolan and, independently, by Juhâsz and Weiss. Just showed the consistency of no thin-very tall sBa's and no very Received by the editors January 24, 1984. 1980 Mathematics Subject Classification. Primary 06E99, 03E35. 'This research partially supported by NSF grant #MCS-8102599. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page