Possible Behaviours of the Reflection Ordering of Stationary Sets

Abstract

If $S,T$ are stationary subsets of a regular uncountable cardinal $\kappa$, we say that $S$ reflects fully in $T$, $S<T$, if for almost all $\alpha \in T$ (except a nonstationary set) $S\cap \alpha$ is stationary in $\alpha .$ This relation is known to be a well founded partial ordering. We say that a given poset $P$ is realized by the reflection ordering if there is a maximal antichain $⟨X_p;p\in P⟩$ of stationary subsets of $Reg(\kappa )$ so that $$\forall p,q\in P\forall S\subseteq X_p,T\subseteq X_q\text{ stationary}:(S<T\leftrightarrow p{<}_{P}q).$$ We prove that if $\kappa$ is $\Cal P _2 κ-$strong and $P$ an arbitrary well founded poset of cardinality $\leq \k^+$ then there is a generic extension where P is realized by the reflection ordering on $\kappa .$