Journal of Symbolic Logic

Some Remarks on Openly Generated Boolean Algebras

Sakae Fuchino

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A Boolean algebra $B$ is said to be openly generated if $\{A: A \leq_{rc} B, |A| = \aleph_0\}$ includes a club subset of $\lbrack B\rbrack^{\aleph_0}$. We show: $(V = L)$. For any cardinal $\kappa$ there exists an $\mathscr{L}_{\infty\kappa}$-free Boolean algebra which is not openly generated (Proposition 4.1). ($MA^+(\sigma$-closed)). Every $\mathscr{L}_{\infty\aleph_a}$-free Boolean algebra is openly generated (Theorem 4.2). The last assertion follows from a characterization of openly generated Boolean algebras under $MA^+(\sigma$-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Scepin [15] (Proposition 4.3 and Theorem 4.4).

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J. Symbolic Logic, Volume 59, Issue 1 (1994), 302-310.

First available in Project Euclid: 6 July 2007

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Fuchino, Sakae. Some Remarks on Openly Generated Boolean Algebras. J. Symbolic Logic 59 (1994), no. 1, 302--310.

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