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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:
 600000 Structure theory of Boolean algebras 300000 Other combinatorial set theory 3e+50 Continuum hypothesis and Martin’s axiom
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