## Journal of Symbolic Logic

### Some Remarks on Openly Generated Boolean Algebras

Sakae Fuchino

#### Abstract

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).

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744451

Mathematical Reviews number (MathSciNet)
MR1264981

Zentralblatt MATH identifier
0803.03031

JSTOR