Skip to main content
SpringerLink
Log in

Downward closure of depth in countable Boolean algebras

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We study the set of depths of relative algebras of countable Boolean algebras, in particular the extent to which this set may not be downward closed within the countable ordinals for a fixed countable Boolean algebra. Doing so, we exhibit a structural difference between the class of arbitrary rank countable Boolean algebras and the class of rank one countable Boolean algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alaev P.E.: Complexity of Boolean algebras and their Scott rank. Algebra Log. 38(643-666), 768 (1999)

    MathSciNet  Google Scholar 

  2. Camerlo R., Gao S.: The completeness of the isomorphism relation for countable Boolean algebras. Trans. Amer. Math. Soc. 353, 491–518 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Heindorf L.: Alternative characterizations of finitary and well-founded Boolean algebras. Algebra Universalis 29, 109–135 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kach A.M.: Depth zero Boolean algebras. Trans. Amer. Math. Soc. 362, 4243–4265 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ketonen J.: The structure of countable Boolean algebras. Ann. of Math. 2(108), 41–89 (1978)

    Article  MathSciNet  Google Scholar 

  6. Marker D.: Model Theory. An Introduction. Graduate Texts in Mathematics, vol. 217. Springer, New York (2002)

  7. Pierce, R.S.: Countable Boolean algebras. In: Handbook of Boolean Algebras, vol. 3, pp. 775-876. North-Holland, Amsterdam (1989)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Asher M. Kach

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Steffen Lempp

Authors
  1. Asher M. Kach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steffen Lempp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Asher M. Kach.

Additional information

Presented by A. Dow.

The first author’s research was partially supported via a Post-Doctoral Fellowship from the Marsden Fund of New Zealand. The second author’s research was supported by NSF Grant DMS-0555381 and Grant # 13407 by the John Templeton Foundation entitled Exploring the Infinite by Finitary Means.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kach, A.M., Lempp, S. Downward closure of depth in countable Boolean algebras. Algebra Univers. 68, 57–74 (2012). https://doi.org/10.1007/s00012-012-0188-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-012-0188-x

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation