Ehrenfeucht-Fraisse games on a class of scattered linear orders

Two structures $A$ and $B$ are $n$-equivalent if player II has a winning strategy in the $n$-move Ehrenfeucht-Fra\"iss\'e game on $A$ and $B$. In earlier papers we studied $n$-equivalence classes of ordinals and coloured ordinals. In this paper we similarly treat a class of scattered order-types, focussing on monomials and sums of monomials in $\omega$ and its reverse $\omega^*$.