Counterexamples in minimally generated Boolean algebras. (English) Zbl 0676.06020

Koppelberg, Sabine

Acta Univ. Carol., Math. Phys. 29, No. 2, 27-36 (1988).

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

Keywords:

diamond; continuum hypothesis; free algebra; minimally generated Boolean algebras; free product; retractive Boolean algebra