Posets of Copies of Countable Scattered Linear Orders

TL;DR

This paper investigates the structure of the poset of isomorphic suborders of countable scattered linear orders, showing it is -closed and atomless, and under CH, all such posets are forcing-equivalent to P()/Fin.

Contribution

It demonstrates that the separative quotient of these posets is -closed and atomless, establishing forcing-equivalence under CH.

Findings

Poset of isomorphic suborders is -closed and atomless

Under CH, all such posets are forcing-equivalent to P()/Fin

Provides structural insight into countable scattered linear orders

Abstract

We show that the separative quotient of the poset (P(L),\subset) of isomorphic suborders of a countable scattered linear order L is \sigma-closed and atomless. So, under the CH, all these posets are forcing-equivalent (to P(\omega)/Fin).