On the Jump Number of Lexicographic Sums of Ordered Sets

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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) + ΣxPs(Qx), where s(X) denotes the jump number of an ordered set X. We first show that w(P) − 1 + ΣxPs(Qx)≤ s(Q) ≤ s(P) + ΣxPs(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.

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