Krull dimension of types in a class of first-order theories

TL;DR

This paper investigates a specific class of first-order theories, demonstrating properties like amalgamation, the existence of a model-companion, strong minimality, and finite Krull dimension of types.

Contribution

It introduces a new class of theories with particular type properties and proves key model-theoretic features such as amalgamation and finite Krull dimension.

Findings

Theories have the amalgamation property.

Existence of a strongly minimal model-companion.

All formulas have finite Krull dimension.

Abstract

We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-companion is strongly minimal. We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension.