# Nonequational Stable Groups

## TL;DR

This paper introduces a combinatorial method to identify non-equational formulas and applies it to prove that free groups and certain free products are non-equational, advancing understanding of their logical structure.

## Contribution

It provides a new combinatorial criterion for non-equationality and extends the non-equationality result to a broader class of free products of groups.

## Key findings

- Proves free groups are non-equational using the new criterion.
- Extends non-equationality to free products of the form G*F_ω.
- Introduces a method for verifying non-equational formulas in group theories.

## Abstract

We introduce a combinatorial criterion for verifying whether a formula is not the conjunction of an equation and a co-equation. Using this, we give a proof for the nonequationality of the free group. Furthermore, we generalize the latter result to the first-order theory of any free product of groups of the form $G*\mathbb{F}_{\omega}$.