## Notre Dame Journal of Formal Logic

### Depth of Boolean Algebras

#### Abstract

Suppose $D$ is an ultrafilter on $\kappa$ and $\lambda^\kappa = \lambda$. We prove that if ${\bf B}_i$ is a Boolean algebra for every $i < \kappa$ and $\lambda$ bounds the depth of every ${\bf B}_i$, then the depth of the ultraproduct of the ${\bf B}_i$'s mod $D$ is bounded by $\lambda^+$. We also show that for singular cardinals with small cofinality, there is no gap at all. This gives a partial answer to a previous problem raised by Monk.

#### Article information

Source
Notre Dame J. Formal Logic Volume 52, Number 3 (2011), 307-314.

Dates
First available in Project Euclid: 28 July 2011

http://projecteuclid.org/euclid.ndjfl/1311875776

Digital Object Identifier
doi:10.1215/00294527-1435474

Mathematical Reviews number (MathSciNet)
MR2822491

Zentralblatt MATH identifier
1141.06010

#### Citation

Garti, Shimon; Shelah, Saharon. Depth of Boolean Algebras. Notre Dame J. Formal Logic 52 (2011), no. 3, 307--314. doi:10.1215/00294527-1435474. http://projecteuclid.org/euclid.ndjfl/1311875776.

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