Atomic saturation of reduced powers

TL;DR

This paper extends theorems on ultra-power saturation to reduced powers, focusing on atomic formulas, and explores the limitations of these generalizations related to SOP_2 and SOP_3 properties.

Contribution

It generalizes saturation results for dense linear orders and SOP_3 pairs to reduced powers, highlighting the boundaries of these extensions.

Findings

Saturation theory for dense linear orders is maximal.

Pairs with SOP_3 are also maximal in this context.

SOP_2 does not allow for similar generalizations.

Abstract

Our aim was to generalize some theorems about the saturation of ultra-powers to reduced powers. Naturally, we deal via saturation for types consisting of atomic formulas. We succeed to generalize the theory of dense linear is maximal and so is any pair (T, Delta) which is SOP_3 (Delta consists of atomic or conjunction of atomic formulas). However, SOP_2 is not enough, so the p equals t theorem cannot be generalized in this case. Similarly the unique dual of cofinality.