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 $\mathrm{pair}(T,\mathrm{\Delta })$ 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 $\mathfrak{p}=\mathfrak{t}$ theorem) cannot be generalized in this case. Similarly the uniqueness of the dual cofinality fails in this context.

中文翻译：

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降低功率的原子饱和

我们的目标是尝试将一些关于超能力饱和到降低能力的定理推广。自然地，我们处理由原子公式组成的类型的饱和。我们成功地概括了“稠密线性有序理论（或具有严格有序性质的T）是极大的，因此任何$\mathrm{一对}(吨,\mathrm{\Delta })$即 SOP ₃ ”，（其中 Δ 由原子式或原子式的组合组成）。然而，关于“处理对称预切割就足够了”的定理（所以$\mathfrak{磷}=\mathfrak{吨}$定理）在这种情况下不能推广。同样，双重共终结性的唯一性在这种情况下也失败了。