Ehrenfeucht-Fraisse games on a class of scattered linear orders

TL;DR

This paper investigates the application of Ehrenfeucht-Fra"issé games to a specific class of scattered linear orders, focusing on monomials and sums involving  and *, to understand their structural equivalences.

Contribution

It extends previous work on ordinals by analyzing $n$-equivalence classes of scattered order-types, specifically monomials and sums in  and *.

Findings

Characterization of $n$-equivalence classes for these scattered orders

Identification of structural properties influencing Ehrenfeucht-Fra"isse9 game outcomes

Extension of ordinal equivalence results to a broader class of scattered orders

Abstract

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 $\omega^*$.