# Saturated Free Algebras and Almost Indiscernible Theories

## TL;DR

This paper generalizes the concept of almost indiscernible theories to uncountable languages, establishing their stability properties and analyzing saturated free algebras, especially in module theories, with new structural theorems and counterexamples.

## Contribution

It extends almost indiscernible theories to uncountable contexts and provides a structure theorem for large models, along with characterizations of saturated free modules and a counterexample to a prior conjecture.

## Key findings

- Almost indiscernible theories are nonmultidimensional, superstable, and stable in all large cardinals.
- Large models are in the algebraic closure of independent sets of weight one types.
- Characterization of rings R for which free R-modules are saturated, with a counterexample to a conjecture.

## Abstract

We extend the concept of "almost indiscernible theory" introduced by Pillay and Sklinos in [Bull. Symb. Log., 2015] (which was itself a modernization and expansion of Baldwin and Shelah [Algebra Universalis, 1983]), to uncountable languages and uncountable parameter sequences. Roughly speaking a theory $T$ is almost indiscernible if some saturated model is in the algebraic closure of an indiscernible set of sequences. We show that such a theory $T$ is nonmultidimensional, superstable, and stable in all cardinals $\ge |T |$ . We prove a structure theorem for sufficiently large $a$-models $M$: Theorem 2.10 which states that over a suitable base, $M$ is in the algebraic closure of an independent set of realizations of weight one types (in possibly infinitely many variables). We also explore further the saturated free algebras of Baldwin and Shelah in both the countable and uncountable context. We study in particular theories and varieties of $R$-modules, characterizing those rings $R$ for which the free $R$-module on $|R|^+$ generators is saturated (Theorem 3.15), and pointing out a counterexample to a conjecture from Pillay-Sklinos (Example 3.16).