More constructions for Boolean algebras

DOI10.1007/S001530100099zbMATH Open1023.03043arXivmath/9605235OpenAlexW2593290804MaRDI QIDQ1407547

Publication date: 16 September 2003

Published in: Archive for Mathematical Logic (Search for Journal in Brave)

Abstract: We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of B_1 and of B_2 than kappa. We give a condition (for lambda, mu and theta) which implies that for some BA A_theta there are B_1=B^1_{lambda, mu, theta} and B_2B^2_{lambda, mu, theta} such that Depth (B_t) <= mu and Depth (B_1 oplus_{A_theta} B_1) >= lambda. We then investigate for a fixed A, the existence of such B_1,B_2 giving sufficient and necessary conditions, involving consistency results. Further we prove that e.g. if B is a BA of cardinality lambda, lambda >= mu and lambda, mu are strong limit singular of the same cofinality, then B has a homomorphic image of cardinality mu (and with mu ultrafilters). Next we show that for a BA B, if d(B)^kappa <|B| then ind (B)> kappa or Depth (B) >= log (|B|). Finally we prove that if square_lambda holds and lambda = lambda^{aleph_0} then for some BAs B_n, Depth (B_n) <= lambda but for any uniform ultrafilter D on omega, prod_{n< omega} B_n/D has depth >= lambda^+ .

Full work available at URL: https://arxiv.org/abs/math/9605235

zbMATH Keywords

forcing Boolean algebras depth cardinal invariants ultraproducts free product pcf application ZFC constructions

Mathematics Subject Classification ID

Consistency and independence results (03E35) Structure theory of Boolean algebras (06E05) Ordered sets and their cofinalities; pcf theory (03E04)