# On the jump number of lexicographic sums of ordered sets

Czechoslovak Mathematical Journal (2003)

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

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topJung, 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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