Order

, Volume 24, Issue 1, pp 59–73 | Cite as

A Half-Space Approach to Order Dimension

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Abstract

The aim of the present paper is to investigate the half-spaces in the convexity structure of all quasiorders on a given set and to use them in an alternative approach to classical order dimension. The main result states that linear orders can almost always be replaced by half-space quasiorders in the definition of the dimension of a partially ordered set.

Keywords

Convexity Quasiorder Preorder Half-space Dimension 

Mathematics Subject Classifications (2000)

06A06 06A07 52A01 

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References

  1. 1.
    Bonnet, R., Pouzet, M.: Linear extensions of ordered sets. In: Rival, I. (ed.) Ordered Sets, Proceedings of the Nato Advanced Study Institute Conference held in Banff, pp. 125–170, August 28–September 12, 1981. Reidel Publishing, Dordrecht (1982)Google Scholar
  2. 2.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63, 600–610 (1941)MATHCrossRefGoogle Scholar
  3. 3.
    Foldes, S.: On intervals in relational structures. Zeitschrift Math. Log. Grund. Math. 26, 97–101 (1980)MATHGoogle Scholar
  4. 4.
    Foldes, S., Radeleczki, S.: On interval decomposition lattices. Discus. Math., Gen. Algebra Appl. 24, 95–114 (2004)MATHGoogle Scholar
  5. 5.
    Hausdorff, F.: Grundzüge einer Theorie der geordneten Mengen. Math. Ann. 65(4), 435–505 (1908)CrossRefGoogle Scholar
  6. 6.
    Körtesi, P. , Radeleczki, S. , Szilágyi, Sz.: Congruences and isotone maps on partially ordered sets. Math. Pannonica 16(1), 39–55 (2005)MATHGoogle Scholar
  7. 7.
    Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fundam. Math. 16, 386–389 (1930)Google Scholar
  8. 8.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets, Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)MATHGoogle Scholar
  9. 9.
    van de Vel, M.L.J.: Theory of Convex Structures, North-Holland Mathematical Library, 50. North-Holland Publishing, Amsterdam (1993)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institute of MathematicsTampere University of TechnologyTampereFinland
  2. 2.Institute of MathematicsUniversity of MiskolcMiskolcHungary

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