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Fractional dimension of partial orders

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Abstract

Given a partially ordered setP=(X, ≤), a collection of linear extensions {L 1,L 2,...,L r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL i (andy<x in someL j ). For a positive integerk, we call a multiset {L 1,L 2,...,L t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask→∞. We prove various results about the fractional dimension of a poset.

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Authors and Affiliations

  1. Department of Statistical and Mathematical Sciences, London School of Economics and Political Science, Houghton Street, WC2A 2AE, London, U.K.

    Graham R. Brightwell

  2. Department of Mathematical Sciences, The Johns Hopkins University, 21218-2689, Baltimore, MD, USA

    Edward R. Scheinerman

Authors
  1. Graham R. Brightwell
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  2. Edward R. Scheinerman
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Additional information

Communicated by H. A. Kierstead

Research supported in part by the Office of Naval Research.

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Brightwell, G.R., Scheinerman, E.R. Fractional dimension of partial orders. Order 9, 139–158 (1992). https://doi.org/10.1007/BF00814406

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