Possible Behaviours of the Reflection Ordering of Stationary Sets

If S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T , S < T , if for almost all α ∈ T (except a nonstationary set) S ∩ α is stationary in α. 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 〈Xp; p ∈ P 〉 of stationary subsets of Reg(κ) so that ∀p, q ∈ P ∀S ⊆ Xp, T ⊆ Xq stationary : (S < T ↔ p <P q). We prove that if V = L[ −→ U ], o (κ) = κ, and P is an arbitrary well-founded poset of cardinality ≤ κ then there is a generic extension where P is realized by the reflection ordering on κ.