Saturated free algebras revisited

TL;DR

This paper revisits Baldwin-Shelah's results on saturated free algebras, providing a general exposition and new insights into forking in saturated free algebra models, with parallels to free group theory.

Contribution

It offers a comprehensive exposition of saturated free algebras and introduces new observations on forking in these models, extending known results to a broader context.

Findings

Description of forking in saturated free algebras

Generalization of Baldwin-Shelah results

New insights analogous to free group theory

Abstract

We give an exposition of results of Baldwin-Shelah on saturated free algebras, at the level of generality of complete first order theories $T$ with a saturated model $M$ which is in the algebraic closure of an indiscernible set. We then make some new observations when $M$ is a saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.