# Atoms in infinite dimensional free sequence-set algebras

## TL;DR

This paper investigates the atomic structure of free algebras in various classes related to cylindric algebras, confirming Tarski's conjecture for some classes and providing counterexamples for others.

## Contribution

It proves Tarski's conjecture holds for certain classes of algebras but shows the free $ ext{Crs}_ ext{alpha}$ algebra is atomless, highlighting differences among algebra classes.

## Key findings

- Tarski's conjecture is true for $ ext{D}_ ext{alpha}$ and $ ext{G}_ ext{alpha}$ classes.
- The free $ ext{Crs}_ ext{alpha}$ algebra is atomless.
- Confirmed the conjecture for classes $ ext{D}_ ext{alpha}$ and $ ext{G}_ ext{alpha}$.

## Abstract

A. Tarski proved that the m-generated free algebra of $\mathrm{CA}_{\alpha}$, the class of cylindric algebras of dimension $\alpha$, contains exactly $2^m$ zero-dimensional atoms, when $m\ge 1$ is a finite cardinal and $\alpha$ is an arbitrary ordinal. He conjectured that, when $\alpha$ is infinite, there are no more atoms. This conjecture has not been confirmed or denied yet. In this article, we show that Tarski's conjecture is true if $\mathrm{CA}_{\alpha}$ is replaced by $\mathrm{D}_{\alpha}$, $\mathrm{G}_{\alpha}$, but the $m$-generated free $\mathrm{Crs}_{\alpha}$ algebra is atomless.