Non-locally modular regular types in classifiable theories

TL;DR

This paper explores properties of regular types in classifiable theories, introducing strong semi-regularity concepts and demonstrating how domination leads to minimal and constructible models under certain conditions.

Contribution

It introduces the notion of strong p-semi-regularity and establishes new results linking domination, isolation, and model minimality in classifiable theories.

Findings

Strong p-semi-regularity implies semi-regularity for non-locally modular types.

Domination by realizations of regular types leads to constructible and minimal models.

Results apply to countable, classifiable theories with regular, non-locally modular types.

Abstract

We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, "domination implies isolation" which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $p\in S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.