On a poset algebra which is hereditarily but not canonically well generated

Matatyahu Rubin, Robert Bonnet

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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(א1) * B(א0) is wellgenerated. (b) B(א1) * B(א1) contains a non well-generated Boolean subalgebra. Canonical well-generatedness is defined in the introduction. Recall that B(א1) * B(א0) is canonically well-generated, and thus well-generated. We prove the following result. THEOREM 2: B(א1) * B(א0) contains a non canonically well generated Boolean subalgebra. In contrast with Theorem 1(b), we have the following result. THEOREM 3: Let A = {aα: α < א1} ⊆ ℘(ω) be a strictly increasing sequence in the relation of almost containment. Let B be the subalgebra of ℘(ω) generated by {{n}: n ∈ א0} ∪ A. Then B is superatomic, and B is not embeddable in a well-generated algebra.

Original languageEnglish
Pages (from-to)299-326
Number of pages28
JournalIsrael Journal of Mathematics
StatePublished - 1 Jan 2003

ASJC Scopus subject areas

  • Mathematics (all)


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