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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
Category: math
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
Date available: 2009-09-24T11:02:06Z
Last updated: 2016-04-07
Stable URL:
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