On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

TL;DR

This paper investigates finitely generated models of countable theories with limited nonisomorphic models, introducing a rank concept to analyze their properties and conditions for primeness, extending group-theoretic properties to model theory.

Contribution

It introduces a rank notion for finitely generated models and characterizes prime models within countable theories, linking model properties to group-theoretic concepts.

Findings

Every finitely generated model has an ordinal rank.

A property analogous to the Hopf property is established for these models.

A criterion for a model to be prime of its theory is provided.

Abstract

We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a prop- erty of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks.