# Sh:652

- Shelah, S. (2002).
*More constructions for Boolean algebras*. Arch. Math. Logic,**41**(5), 401–441. arXiv: math/9605235 DOI: 10.1007/s001530100099 MR: 1918108 -
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)\leq\mu and Depth(B_1\oplus_{A_\theta} B_1) \geq \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\ge\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)\geq\log(|B|). Finally we prove that if \square_\lambda holds and \lambda=\lambda^{\aleph_0} then for some BAs B_n, Depth(B_n)\leq\lambda but for any uniform ultrafilter D on \omega, \prod_{n<\omega} B_n/D has depth \ge\lambda^+. - published version (41p)

Bib entry

@article{Sh:652, author = {Shelah, Saharon}, title = {{More constructions for Boolean algebras}}, journal = {Arch. Math. Logic}, fjournal = {Archive for Mathematical Logic}, volume = {41}, number = {5}, year = {2002}, pages = {401--441}, issn = {0933-5846}, doi = {10.1007/s001530100099}, mrclass = {03E04 (03G05 06E05)}, mrnumber = {1918108}, mrreviewer = {Klaas Pieter Hart}, doi = {10.1007/s001530100099}, note = {\href{https://arxiv.org/abs/math/9605235}{arXiv: math/9605235}}, arxiv_number = {math/9605235} }