Depth of boolean algebras

Shimon Garti; Saharon Shela

doi:10.1215/00294527-1435474

Depth of boolean algebras

Shimon Garti^*, Saharon Shela

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Suppose D is an ultrafilter on κ and λ^κ = λ. We prove that if B_i is a Boolean algebra for every i < κ and λ bounds the depth of every B_i , then the depth of the ultraproduct of the B_i 's mod D is bounded by λ⁺. 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.

Original language	English
Pages (from-to)	307-314
Number of pages	8
Journal	Notre Dame Journal of Formal Logic
Volume	52
Issue number	3
DOIs	https://doi.org/10.1215/00294527-1435474
State	Published - 2011

Keywords

Boolean algebras
Constructability
Depth

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10.1215/00294527-1435474

Cite this

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abstract = "Suppose D is an ultrafilter on κ and λκ = λ. We prove that if Bi is a Boolean algebra for every i < κ and λ bounds the depth of every Bi , then the depth of the ultraproduct of the Bi 's mod D is bounded by λ+. 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.",

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AB - Suppose D is an ultrafilter on κ and λκ = λ. We prove that if Bi is a Boolean algebra for every i < κ and λ bounds the depth of every Bi , then the depth of the ultraproduct of the Bi 's mod D is bounded by λ+. 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.

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KW - Constructability

KW - Depth

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Depth of boolean algebras

Abstract

Keywords

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