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, Volume 10, Issue 3, pp 227–237 | Cite as

Spherical containment and the Minkowski dimension of partial orders

  • David A. Meyer
Article

Abstract

The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresSd−2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.

Mathematics Subject Classifications (1991)

06A06 52A37 

Key words

Causal order spherical (containment) order Minkowski dimension 

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • David A. Meyer
    • 1
  1. 1.Department of PhysicsSyracuse UniversitySyracuseU.S.A.

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