Distorted sums of models

TL;DR

This paper explores distorted sums of models, focusing on types and local types, simplifying previous proofs, and extending results to models with distant functions, including an improved Gaifman theorem.

Contribution

It simplifies proofs and bounds related to types in distorted sums and extends Gaifman's theorem to models with distant functions.

Findings

Type of a sequence is determined by local type.

Simplified proofs and improved bounds on radii.

Extended Gaifman theorem to models with distant functions.

Abstract

Distorted sums of models were introduced and discussed in [Sh:463]. This notion generalizes the notion of disjoint (or direct) sums of models by letting the summands overlap. In the first section we investigate types in distorted sums and show that the type of a sequence of elements A is determined by the `local' type of A (i.e. the type restricted to a neighborhood of A). We simplify the proofs in [Sh:463] and improve the bounds on the radii needed to determine the types. Natural examples of distorted sums are models with distant functions. In the second and third sections we discuss such models and improve a theorem by Gaifman, that states that each formula is equivalent to a boolean combination of local formulas.