Theories with few non-algebraic types over models, and their decompositions

TL;DR

This paper explores the relationship between model decompositions and mutual algebraicity, establishing that certain bounded decompositions are equivalent to mutual algebraicity and characterizing theories with uniform bounds on types.

Contribution

It demonstrates the equivalence between model decompositions and mutual algebraicity, providing new characterizations of theories with bounded types.

Findings

Decompositions of models are equivalent to mutual algebraicity.

Mutual algebraicity characterized by uniform bounds on types.

Theories with bounded types have specific structural properties.

Abstract

We consider several ways of decomposing models into parts of bounded size forming a congruence over a base, and show that admitting any such decomposition is equivalent to mutual algebraicity at the level of theories. We also show that a theory $T$ is mutually algebraic if and only if there is a uniform bound on the number of coordinate-wise non-algebraic types over every model, regardless of its cardinality.