Foundations of Physics

, Volume 20, Issue 6, pp 715–732 | Cite as

Interval order and semiorder lattices

  • M. F. Janowitz
Part II. Invited Papers Dedicated To The Memory Of Charles H. Randall (1928–1987)

Abstract

When a set of closed intervals of the reals is partially ordered by decreeing that A<B when A lies strictly to the left of B, the resulting structure is called an interval order. Semiorders may be viewed as interval orders that arise from closed intervals having a fixed length. The paper initiates a careful study of interval orders and semiorders that happen also to be lattices. A structure theory is obtained for a class of interval order lattices that includes all such lattices of finite length. Characterizations are given of when these lattices are modular or distributive, as well as when they are semiorders. The theory is of some interest because the completion by cuts of an interval order is necessarily an interval order lattice. Though it is shown that the completion by cuts of a semiorder need not be a semiorder, necessary and sufficient conditions are given for a lattice of finite length to be isomorphic to the completion by cuts of a semiorder.

Keywords

Closed Interval Structure Theory Careful Study Finite Length Interval Order 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. E. Adams and M. Gould, “Finite semilattices whose monoids of endomorphisms are regular,” to appear inTrans. Amer. Math. Soc. Google Scholar
  2. 2.
    P. C. Fishburn,J. Math. Psych. 7, 144 (1970).CrossRefGoogle Scholar
  3. 3.
    P. C. Fishburn,Interval Orders and Interval Graphs (Wiley-Interscience, New York, 1985).Google Scholar
  4. 4.
    R. D. Luce,Econometrica 24, 178 (1956).Google Scholar
  5. 5.
    F. Maeda and S. Maeda,Theory of Symmetric Lattices (Springer, Berlin, 1970).Google Scholar
  6. 6.
    B. Monjardet,Order 5, 211 (1988).Google Scholar
  7. 7.
    F. Roberts,Measurement Theory, Encyclopedia of Mathematics and Its Applications, Vol. 7) (Addison-Wesley, Reading, Massachusetts, 1979).Google Scholar
  8. 8.
    N. Wiener,Proc. Camb. Philos. Soc. 17, 441 (1914).Google Scholar
  9. 9.
    N. Wiener,Proc. Camb. Philos. Soc. 18, 14 (1915).Google Scholar
  10. 10.
    N. Wiener,Proc. London Math. Soc. 19, 181 (1921).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • M. F. Janowitz
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of Massachusetts at Amherst, Lederle Graduate Research CenterAmherst

Personalised recommendations