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A characterization of complete atomic Boolean algebra.

Francesc Esteva (1977)


In this note we give a characterization of complete atomic Boolean algebras by means of complete atomic lattices. We find that unicity of the representation of the maximum as union of atoms and Lambda-infinite distributivity law are necessary and sufficient conditions for the lattice to be a complete atomic Boolean algebra.

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

Miloš S. Kurilić, Aleksandar Pavlović (2014)

Czechoslovak Mathematical Journal

We compare the forcing-related properties of a complete Boolean algebra 𝔹 with the properties of the convergences λ s (the algebraic convergence) and λ ls on 𝔹 generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λ ls is a topological convergence iff forcing by 𝔹 does not produce new reals and that λ ls is weakly topological if 𝔹 satisfies condition ( ) (implied by the 𝔱 -cc). On the other hand, if λ ls is a weakly topological convergence, then 𝔹 is a 2 𝔥 -cc algebra...

Chains and antichains in Boolean algebras

M. Losada, Stevo Todorčević (2000)

Fundamenta Mathematicae

We give an affirmative answer to problem DJ from Fremlin’s list [8] which asks whether M A ω 1 implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.

Coherent ultrafilters and nonhomogeneity

Jan Starý (2015)

Commentationes Mathematicae Universitatis Carolinae

We introduce the notion of a coherent P -ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a P -point on ω , and show that these ultrafilters exist generically under 𝔠 = 𝔡 . This improves the known existence result of Ketonen [On the existence of P -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1, 233–241] can...

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