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# On Some Small Cardinals for Boolean Algebras

Ralph McKenzie and J. Donald Monk
The Journal of Symbolic Logic
Vol. 69, No. 3 (Sep., 2004), pp. 674-682
Stable URL: http://www.jstor.org/stable/30041752
Page Count: 9
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## Abstract

Assume that all algebras are atomless. (1) $Spind(A x B) = Spind(A) \cup Spind(B)$. (2) $(\prod_{i\inI}^{w} = {\omega} \cup \bigcup_{i\inI}$ $Spind(A_{i})$. Now suppose that $\kappa$ and $\lambda$ are infinite cardinals, with $kappa$ uncountable and regular and with $\kappa \textless \lambda$. (3) There is an atomless Boolean algebra A such that $\mathfrak{u}(A) = \kappa$ and $i(A) = \lambda$. (4) If $\lambda$ is also regular, then there is an atomless Boolean algebra A such that $t(A) = \mathfrak{s}(A) = \kappa$ and $\mathfrak{a}(A) = \lambda$. All results are in ZFC, and answer some problems posed in Monk [01] and Monk [$\infty$].

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