On the Jump Number of Lexicographic Sums of Ordered Sets
Hyung Chan Jung, Jeh Gwon Lee · Czechoslovak Mathematical Journal · 2003
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, $$s\left( Q \right) = s\left( P \right) + \sum\limits_{x \in P} {s\left( {Qx} \right)} $$ , where s(X) denotes the jump number of an ordered set X. We first show that $$w\left( P \right) - 1 + \sum\limits_{x \in P} {s\left( {Qx} \right)} \leqslant {\text{ }}s\left( Q \right) \leqslant s\left( P \right) + \sum\limits_{x \in P} {s\left( {Qx} \right)} $$ 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.