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.
In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as $R[x]$-modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.