Czechoslovak Mathematical Journal. 53(2):343–349, JUNE 2003

Issn Print: 0011-4642

Publication Date: June 2003

# On the Jump Number of Lexicographic Sums of Ordered Sets

Hyung Jung;Jeh Lee;

+ Author Information

1Liberal Arts and Sciences, Korea University of Technology and Education, Byungchon, Chonan, Chungnam, 330-860, Korea, hcjung@kut.kite.ac.kr2Department of Mathematics, Sogang University, Seoul, 121-742, Korea, ljg@sogang.ac.kr

### Abstract

Let Q be the lexicographic sum of finite ordered sets Qx 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 Qx and P, that is, s(Q)=s(P) + Σx ∈ Ps(Qx), where s(X) denotes the jump number of an ordered set X. We first show that w(P) − 1 + Σx ∈ Ps(Qx)≤ s(Q) ≤ s(P) + Σx ∈ Ps(Qx), 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.