The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties

TL;DR

This paper studies the category of atomic monoids, establishing its completeness and cocompleteness, and explores how product and coproduct operations affect key factorization invariants.

Contribution

It introduces the category of atomic monoids, computes all limits and colimits, and analyzes arithmetic properties related to factorization invariants.

Findings

The category of atomic monoids is complete and cocomplete.

Explicit formulas for invariants related to factorization lengths.

Analysis of arithmetic properties of products and coproducts.

Abstract

We introduce and investigate the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving monoid homomorphisms, which is a (non-full) subcategory of the usual category of monoids. In particular, we compute all limits and colimits, showing that $\mathsf{AtoMon}$ is a complete and cocomplete category. We also address certain arithmetic properties of products and coproducts, providing explicit formulas for some fundamental invariants associated with factorization lengths in atomic monoids.