Fractional dimension of partial orders |
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Authors: | Graham R Brightwell Edward R Scheinerman |
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Institution: | (1) Department of Statistical and Mathematical Sciences, London School of Economics and Political Science, Houghton Street, WC2A 2AE London, U.K.;(2) Department of Mathematical Sciences, The Johns Hopkins University, 21218-2689 Baltimore, MD, USA |
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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![rarr](/content/mn263t1q74w14134/xxlarge8594.gif) . We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research. |
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Keywords: | 06A06 |
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