Atomic saturation of reduced powers

Our aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize “the theory of dense linear order (or T with the strict order property) is maximal and so is any (Formula presented.) which is SOP₃”, (where Δ consists of atomic or conjunction of atomic formulas). However, the theorem on “it is enough to deal with symmetric pre-cuts” (so the (Formula presented.) theorem) cannot be generalized in this case. Similarly the uniqueness of the dual cofinality fails in this context.