# Sh:620

- Shelah, S. (1999).
*Special subsets of ^{\mathrm{cf}(\mu)}\mu, Boolean algebras and Maharam measure algebras*. Topology Appl.,**99**(2-3), 135–235. arXiv: math/9804156 DOI: 10.1016/S0166-8641(99)00138-8 MR: 1728851 -
Abstract:

The original theme of the paper is the existence proof of “there is \bar{\eta}=\langle\eta_\alpha:\alpha<\lambda\rangle which is a (\lambda,J)-sequence for \bar{I}=\langle I_i:i< \delta\rangle, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and Sierpinski sets, but for the product \prod_{i< \delta} Dom(I_i), the existence proofs are related to pcf . The second theme is when does a Boolean algebra B has free caliber \lambda (i.e. if X\subseteq B and |X|=\lambda, then for some Y\subseteq X with |Y|=\lambda and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and \kappa-cc Boolean algebras. A central case \lambda= (\beth_\omega)^+ or more generally, \lambda=\mu^+ for \mu strong limit singular of “small” cofinality. A second one is \mu=\mu^{<\kappa}<\lambda< 2^\mu; the main case is \lambda regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) Length, etc. - published version (101p)

Bib entry

@article{Sh:620, author = {Shelah, Saharon}, title = {{Special subsets of $^{\mathrm{cf}(\mu)}\mu$, Boolean algebras and Maharam measure algebras}}, journal = {Topology Appl.}, fjournal = {Topology and its Applications}, volume = {99}, number = {2-3}, year = {1999}, pages = {135--235}, issn = {0166-8641}, mrnumber = {1728851}, mrclass = {03E05 (03E04 03E35 03E75 06E05 28A60)}, doi = {10.1016/S0166-8641(99)00138-8}, note = {\href{https://arxiv.org/abs/math/9804156}{arXiv: math/9804156}}, arxiv_number = {math/9804156}, conference = {8th Prague Topological Symposium on General Topology and its Relations to Modern Analysis and Algebra, Part II (1996)} }