# Factorization invariants of Puiseux monoids generated by geometric sequences

## TL;DR

This paper investigates the factorization properties of Puiseux monoids generated by geometric sequences, revealing their structure and comparing them with numerical monoids from arithmetic sequences, with detailed invariants analysis.

## Contribution

It provides a detailed description of factorization invariants for Puiseux monoids generated by geometric sequences, including sets of lengths and elasticity, expanding understanding beyond numerical monoids.

## Key findings

- All sets of lengths are arithmetic sequences with common difference |a-b|.
- Explicit formulas for elasticity and tameness of S_r.
- Comparison with numerical monoids highlights unique factorization properties.

## Abstract

We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study consists of all atomic monoids of the form $S_r := \langle r^n \mid n \in \mathbb{N}_0 \rangle,$ where $r$ is a positive rational. As the atomic monoids $S_r$ are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of $S_r$ is that all its sets of lengths are arithmetic sequences of the same distance, namely $|a-b|$, where $a,b \in \mathbb{N}$ are such that $r = a/b$ and $\text{gcd}(a,b) = 1$. We prove this, and then use it to study the elasticity and tameness of $S_r$.