# On the jump number of lexicographic sums of ordered sets

• Volume: 53, Issue: 2, page 343-349
• ISSN: 0011-4642

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## Abstract

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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, $s\left(Q\right)=s\left(P\right)+{\sum }_{x\in P}s\left({Q}_{x}\right)$, where $s\left(X\right)$ denotes the jump number of an ordered set $X$. We first show that $w\left(P\right)-1+{\sum }_{x\in P}s\left({Q}_{x}\right)\le s\left(Q\right)\le s\left(P\right)+{\sum }_{x\in P}s\left({Q}_{x}\right)$, where $w\left(X\right)$ denotes the width of an ordered set $X$. Consequently, if $P$ is a Dilworth ordered set, that is, $s\left(P\right)=w\left(P\right)-1$, 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.

## How to cite

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Jung, Hyung Chan, and Lee, Jeh Gwon. "On the jump number of lexicographic sums of ordered sets." Czechoslovak Mathematical Journal 53.2 (2003): 343-349. <http://eudml.org/doc/30781>.

@article{Jung2003,
abstract = {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.},
author = {Jung, Hyung Chan, Lee, Jeh Gwon},
journal = {Czechoslovak Mathematical Journal},
keywords = {ordered set; jump (setup) number; lexicographic sum; jump-critical; ordered set; jump number; lexicographic sum; jump-critical set},
language = {eng},
number = {2},
pages = {343-349},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the jump number of lexicographic sums of ordered sets},
url = {http://eudml.org/doc/30781},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Jung, Hyung Chan
AU - Lee, Jeh Gwon
TI - On the jump number of lexicographic sums of ordered sets
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 2
SP - 343
EP - 349
AB - 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.
LA - eng
KW - ordered set; jump (setup) number; lexicographic sum; jump-critical; ordered set; jump number; lexicographic sum; jump-critical set
UR - http://eudml.org/doc/30781
ER -

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