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Title: On the jump number of lexicographic sums of ordered sets (English)
Author: Jung, Hyung Chan
Author: Lee, Jeh Gwon
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 53
Issue: 2
Year: 2003
Pages: 343-349
Summary lang: English
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Category: math
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Summary: Let $Q$ be the lexicographic sum of finite ordered sets $Q_x$ over a finite ordered set $P$. For some $P$ we can give a formula for the jump number of $Q$ in terms of the jump numbers of $Q_x$ and $P$, that is, $s(Q)=s(P)+ \sum _{x\in P} s(Q_x)$, where $s(X)$ denotes the jump number of an ordered set $X$. We first show that $w(P)-1+\sum _{x\in P} s(Q_x)\le s(Q) \le s(P)+ \sum _{x\in P} s(Q_x)$, where $w(X)$ denotes the width of an ordered set $X$. Consequently, if $P$ is a Dilworth ordered set, that is, $s(P) = w(P)-1$, then the formula holds. We also show that it holds again if $P$ is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical. (English)
Keyword: ordered set
Keyword: jump (setup) number
Keyword: lexicographic sum
Keyword: jump-critical
MSC: 06A07
idZBL: Zbl 1024.06001
idMR: MR1983456
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Date available: 2009-09-24T11:02:06Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/127804
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Reference: [2] R. P.  Dilworth: A decomposition theorem for partially ordered sets.Ann. Math. 51 (1950), 161–166. Zbl 0038.02003, MR 0032578, 10.2307/1969503
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Reference: [4] M. H.  ElZahar and J. H.  Schmerl: On the size of jump-critical ordered sets.Order 1 (1984), 3–5. MR 0745584, 10.1007/BF00396268
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Reference: [7] H. C.  Jung: On the products of some posets: jump number, greediness.Ars Combin. 40 (1995), 109–120. MR 1352773
Reference: [8] E.  Szpilrajn: Sur l’extension de l’ordre partiel.Fund. Math. 16 (1930), 386–389.
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