Skip to main content
SpringerLink
Log in

Simple Inductive Proofs of the Fishburn and Mirkin Theorem and the Scott–Suppes Theorem

  • Published:
Order Aims and scope Submit manuscript

Abstract

This paper presents new proofs of two classic characterization theorems for families of ordered sets. The first is that any finite poset with no restriction isomorphic to \(\underline 2 + \underline 2 \) has an interval representation. The second is that any finite poset with no restriction isomorphic to \(\underline 2 + \underline 2 \) or to \(\underline 3 + \underline 1 \) has a unit interval representation. Both proofs are straightforward and inductive.

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. Trotter, W. T.: Combinatorics and Partially Ordered Sets, Johns Hopkins University Press, Baltimore, Maryland, 1992, p. 3.

    Google Scholar 

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

    Google Scholar 

  3. Mirkin, B. G.: Description of some relations on the set of real-line intervals, J. Math. Psychol. 9 (1972), 243–252.

    Google Scholar 

  4. Mirkin, B. G.: Ob odnom klasse otnoshenij predpochtenija, Matematitcheskije woprosy formirovanija economitcheskich modelei (1970).

  5. Bogart, K. P.: An obvious proof of Fishburn's interval order theorem, Discrete Math. 118 (1993),239–242.

    Google Scholar 

  6. Scott, D. and Suppes, P.: Foundational aspects of theories of measurement, J. Symbol. Logic 23 (1958), 113–128.

    Google Scholar 

  7. Bogart, K. P. and West, D. B.: A short proof that ‘proper = unit’, Discrete Math. 201 (1999), 21–23.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Dartmouth College, Hanover, NH, 03755, U.S.A.

    Barry Balof & Kenneth Bogart

Authors
  1. Barry Balof
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenneth Bogart
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Balof, B., Bogart, K. Simple Inductive Proofs of the Fishburn and Mirkin Theorem and the Scott–Suppes Theorem. Order 20, 49–51 (2003). https://doi.org/10.1023/A:1024430208672

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024430208672

Search

Navigation