### On the jump number of lexicographic sums of ordered sets

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, $s\left(Q\right)=s\left(P\right)+{\sum}_{x\in P}s\left({Q}_{x}\right)$, where $s\left(X\right)$ denotes the jump number of an ordered set $X$. We first show that $w\left(P\right)-1+{\sum}_{x\in P}s\left({Q}_{x}\right)\le s\left(Q\right)\le s\left(P\right)+{\sum}_{x\in P}s\left({Q}_{x}\right)$, where $w\left(X\right)$ denotes the width of an ordered set $X$. Consequently, if $P$ is a Dilworth ordered set, that is, $s\left(P\right)=w\left(P\right)-1$, then the formula holds. We also show that it holds again if $P$ is bipartite. Finally, we prove that the lexicographic sum of...