Koppelberg, Sabine Minimally generated Boolean algebras. (English) Zbl 0676.06019 Order 5, No. 4, 393-406 (1989). 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 \(B_{\alpha}\). This paper proves basic theorems about minimally generated algebras. For example: Theorem. Interval algebras and superatomic algebras are minimally generated. Theorem. Minimally generated algebras are closed under product, homomorphic image, and products of finitely many factors. Theorem. Every minimally generated algebra is co-absolute with an interval algebra. Theorem. A minimally generated algebra cannot have an uncountable free subalgebra. Reviewer: J.Roitman Cited in 2 ReviewsCited in 143 Documents MSC: 06E05 Structure theory of Boolean algebras 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) Keywords:minimally generated Boolean algebras; minimal extension; superatomic algebras; interval algebra PDFBibTeX XMLCite \textit{S. Koppelberg}, Order 5, No. 4, 393--406 (1989; Zbl 0676.06019) Full Text: DOI References: [1] J. Baumgartner (1976) Almost disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9, 401-439. · Zbl 0339.04003 · doi:10.1016/0003-4843(76)90018-8 [2] G. Brenner (1983) A simple construction for rigid and weakly homogeneous Boolean algebras answering a question of Rubin, Proc. Amer. Math. Soc. 87, 601-606. · Zbl 0524.06021 · doi:10.1090/S0002-9939-1983-0687625-3 [3] Th. Jech (1978) Set Theory, Academic Press, New York. · Zbl 0419.03028 [4] S. Koppelberg (1989a) General theory of Boolean algebras, in Handbook of Boolean Algebras (ed. J. D. Monk), North Holland, Amsterdam. · Zbl 0676.06019 [5] S. Koppelberg (1989b) Projective Boolean algebras, in Handbook of Boolean Algebras (ed. J. D. Monk), North Holland, Amsterdam, pp. 741-773. [6] S. Koppelberg (1988) Counterexamples in minimally generated Boolean algebras, Acta Universitatis Carolinae ? Math. Phys. 29(2), 27-36. · Zbl 0676.06020 [7] W. Mitchell (1972) Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5, 21-46. · Zbl 0255.02069 · doi:10.1016/0003-4843(72)90017-4 [8] S. Shelah (1981) On uncountable Boolean algebras with no uncountable pairwise comparable or incomparable sets of elements, Notre Dame J. Formal Logic 22, 301-308. · Zbl 0472.03042 · doi:10.1305/ndjfl/1093883511 [9] S. Todor?evi? (1984) Trees and linearly ordered sets, in Handbook of Set-Theoretic Topology (ed. K. Kunen and J. E. Vaughan), pp. 235-293. · Zbl 0557.54021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.