Spherical containment and the Minkowski dimension of partial orders
 David A. Meyer
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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 spheresS ^{ d−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.
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 Title
 Spherical containment and the Minkowski dimension of partial orders
 Journal

Order
Volume 10, Issue 3 , pp 227237
 Cover Date
 19930901
 DOI
 10.1007/BF01110544
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 06A06
 52A37
 Causal order
 spherical (containment) order
 Minkowski dimension
 Industry Sectors
 Authors

 David A. Meyer ^{(1)}
 Author Affiliations

 1. Department of Physics, Syracuse University, 132441130, Syracuse, NY, U.S.A.