## Abstract

Let B be a superatomic Boolean algebra. B is well generated, if it has a well founded sublattice L such that L generates B. The free product of Boolean algebras B and C is denoted by B * C. If C is a chain then B(C) denotes the interval algebra over C. THEOREM 1: (a) Every Boolean subaJgebra of B(א_{1}) * B(א_{0}) is wellgenerated. (b) B(א_{1}) * B(א_{1}) contains a non well-generated Boolean subalgebra. Canonical well-generatedness is defined in the introduction. Recall that B(א_{1}) * B(א_{0}) is canonically well-generated, and thus well-generated. We prove the following result. THEOREM 2: B(א_{1}) * B(א_{0}) contains a non canonically well generated Boolean subalgebra. In contrast with Theorem 1(b), we have the following result. THEOREM 3: Let A = {a_{α}: α < א_{1}} ⊆ ℘(ω) be a strictly increasing sequence in the relation of almost containment. Let B be the subalgebra of ℘(ω) generated by {{n}: n ∈ א_{0}} ∪ A. Then B is superatomic, and B is not embeddable in a well-generated algebra.

Original language | English |
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Pages (from-to) | 299-326 |

Number of pages | 28 |

Journal | Israel Journal of Mathematics |

Volume | 135 |

DOIs | |

State | Published - 1 Jan 2003 |

## ASJC Scopus subject areas

- Mathematics (all)