# Article

 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: http://hdl.handle.net/10338.dmlcz/127804 . Reference: [1] M.  Chein and M.  Habib: The jump number of dags and posets: an introduction.Ann. Discrete Math. 9 (1980), 189–194. MR 0597371, 10.1016/S0167-5060(08)70060-8 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 Reference: [3] M. H.  ElZahar and I.  Rival: Examples of jump-critical ordered sets.SIAM J.  Alg. Disc. Math. 6 (1985), 713–720. MR 0801000, 10.1137/0606069 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 Reference: [5] M.  Habib: Comparability invariants.Ann. Discrete Math. 23 (1984), 371–386. Zbl 0561.05046, MR 0779861 Reference: [6] M.  Habib and R. H.  Möhring: On some complexity properties of $N$-free posets with bounded decomposition diameter.Discrete Math. 63 (1987), 157–182. MR 0885495, 10.1016/0012-365X(87)90006-9 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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