### Depth of Boolean Algebras

Shimon Garti and Saharon Shelah
Source: Notre Dame J. Formal Logic Volume 52, Number 3 (2011), 307-314.

#### 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.

First Page:
Primary Subjects: 06E05, 03G05
Secondary Subjects: 03E45
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1311875776
Digital Object Identifier: doi:10.1215/00294527-1435474
Zentralblatt MATH identifier: 1141.06010
Mathematical Reviews number (MathSciNet): MR2822491

### References

[1] Chang, C. C., and H. J. Keisler, Model Theory, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1973.
Mathematical Reviews (MathSciNet): MR0409165
Zentralblatt MATH: 0276.02032
[2] Donder, H.-D., "Regularity of ultrafilters and the core model", Israel Journal of Mathematics, vol. 63 (1988), pp. 289–322.
Mathematical Reviews (MathSciNet): MR969944
Zentralblatt MATH: 0663.03037
Digital Object Identifier: doi:10.1007/BF02778036
[3] Garti, S., and S. Shelah, "On Depth and Depth$^+$" of Boolean algebras, Algebra Universalis, vol. 58 (2008), pp. 243–48.
Mathematical Reviews (MathSciNet): MR2386531
Zentralblatt MATH: 1141.06010
Digital Object Identifier: doi:10.1007/s00012-008-2065-1
[4] Garti, S., and S. Shelah, "($\kappa-\theta$)-weak normality", forthcoming in Journal of the Mathematical Society of Japan, (2011).
[5] Monk, J. D., Cardinal Functions on Boolean Algebras, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1990.
Mathematical Reviews (MathSciNet): MR1077622
Zentralblatt MATH: 0706.06009
[6] Monk, J. D., Cardinal Invariants on Boolean Algebras, vol. 142 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1996.
Mathematical Reviews (MathSciNet): MR1393943
Zentralblatt MATH: 0849.03038
[7] Shelah, S., "Products of regular cardinals and cardinal invariants of products of Boolean algebras", Israel Journal of Mathematics, vol. 70 (1990), pp. 129–87.
Mathematical Reviews (MathSciNet): MR1070264
Zentralblatt MATH: 0722.03038
Digital Object Identifier: doi:10.1007/BF02807866
[8] Shelah, S., "Applications of PCF" theory, The Journal of Symbolic Logic, vol. 65 (2000), pp. 1624–74.
Mathematical Reviews (MathSciNet): MR1812172
Zentralblatt MATH: 0981.03048
Digital Object Identifier: doi:10.2307/2695067
[9] Shelah, S., "More constructions for Boolean algebras", Archive for Mathematical Logic, vol. 41 (2002), pp. 401–41.
Mathematical Reviews (MathSciNet): MR1918108
Zentralblatt MATH: 1023.03043
Digital Object Identifier: doi:10.1007/s001530100099
[10] Shelah, S., "The depth of ultraproducts of Boolean algebras", Algebra Universalis, vol. 54 (2005), pp. 91–96.
Mathematical Reviews (MathSciNet): MR2217966
Zentralblatt MATH: 1098.03059
Digital Object Identifier: doi:10.1007/s00012-005-1925-1