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Counterexamples in minimally generated Boolean algebras. (English) Zbl 0676.06020
A Boolean algebra is minimally generated iff it is the union of a continuous well-ordered chain of subalgebras, where each \(B_{\alpha +1}\) is minimally generated over \(B_{\alpha}.\)
Theorem. There is a Boolean algebra which is not minimally generated in which no uncountable free algebra embeds.
Theorem. The class of minimally generated Boolean algebras is not closed under finite free product.
Theorem. Assume \(\diamond\). There is a minimally generated Boolean algebra of cofinality \(\omega_ 1.\)
Theorem. Assume CH. There is a retractive Boolean algebra which is not minimally generated.
Reviewer: J.Roitman

MSC:
06E05 Structure theory of Boolean algebras
03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
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