Skip to main content
SpringerLink
Log in

Infinite Saturated Orders

  • Published:
Order Aims and scope Submit manuscript

Abstract

We generalize the notion of saturated orders to infinite partial orders and give both a set-theoretic and an algebraic characterization of such orders. We then study the proof theoretic strength of the equivalence of these characterizations in the context of reverse mathematics, showing that depending on one’s choice of definitions, this equivalence is either provable in RCA 0 or equivalent to ACA 0.

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. Doignon, J.-P., Falmagne, J.-C.: Knowledge Spaces. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  2. Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fishburn, P.C.: Interval orders and interval graphs. In: Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1985) [A Study of Partially Ordered Sets, A Wiley-Interscience Publication]

    Google Scholar 

  4. Marcone, A.: Interval orders and reverse mathematics. Notre Dame J. Form. Log. 48(3), 425–448 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mirkin, B.G.: Ob odnom klasse otnoshenij predpochtenija. In: Matematitcheskije Woprosy Formirovanija Economitcheskich Modelei. Novosibirsk (1970)

  6. Simpson, S.G.: Subsystems of second order arithmetic. In: Perspectives in Mathematical Logic. Springer, Berlin (1999)

    Google Scholar 

  7. Suck, R.: Parsimonious set representations of orders, a generalization of the interval order concept, and knowledge spaces. Discrete Appl. Math. 127(2), 373–386 (2003) [The 1998 Conference on Ordinal and Symbolic Data Analysis (OSDA ’98) (Amherst, MA)]

    Article  MATH  MathSciNet  Google Scholar 

  8. Suck, R.: Set representations of orders and a structural equivalent of saturation. J. Math. Psychol. 48(3), 159–166 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Chicago, Chicago, IL, USA

    Damir D. Dzhafarov

Authors
  1. Damir D. Dzhafarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Damir D. Dzhafarov.

Additional information

The author was partially supported by an NSF Graduate Research Fellowship. He is grateful to R. Suck for introducing him to this problem, and to E. Dzhafarov, D. Hirschfeldt, A. Montalbán, and R. Soare for valuable comments. The author also thanks the two anonymous referees for suggestions and corrections that helped to improve this work.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dzhafarov, D.D. Infinite Saturated Orders. Order 28, 163–172 (2011). https://doi.org/10.1007/s11083-010-9160-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-010-9160-6

Keywords

Search

Navigation