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The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties

Federico Campanini, Laura Cossu, Salvatore Tringali

TL;DR

This work builds a categorical framework for factorization theory by introducing the category $\mathsf{AtoMon}$ of atomic monoids with atom-preserving morphisms. It proves that $\mathsf{AtoMon}$ is complete and cocomplete, with coproducts realized as free products (coinciding with the Mon coproducts) and a distinct product construction given by submonoids generated by units and atoms. The authors provide explicit, computable descriptions of length-set invariants for these (co)limits, including detailed constructions of equalizers and pullbacks, and show that colimits behave like those in $\mathsf{Mon}$ while limits require dedicated descriptions. These results offer new categorical tools for factorization theory and open avenues for applications to length-set realization problems and the study of transfer-type morphisms, as well as foundations for future work on factorable monoids and premonoid categories. Overall, the paper establishes a robust, concrete link between factorization invariants and universal constructions in a principled categorical setting.

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.

The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties

TL;DR

This work builds a categorical framework for factorization theory by introducing the category of atomic monoids with atom-preserving morphisms. It proves that is complete and cocomplete, with coproducts realized as free products (coinciding with the Mon coproducts) and a distinct product construction given by submonoids generated by units and atoms. The authors provide explicit, computable descriptions of length-set invariants for these (co)limits, including detailed constructions of equalizers and pullbacks, and show that colimits behave like those in while limits require dedicated descriptions. These results offer new categorical tools for factorization theory and open avenues for applications to length-set realization problems and the study of transfer-type morphisms, as well as foundations for future work on factorable monoids and premonoid categories. Overall, the paper establishes a robust, concrete link between factorization invariants and universal constructions in a principled categorical setting.

Abstract

We introduce and investigate the category 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 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.

Paper Structure

This paper contains 14 sections, 23 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities
  4. Monoids.
  5. Free monoids
  6. Factorizations and length sets
  7. Categories
  8. The category of atomic monoids
  9. Coproducts
  10. Products
  11. Other limits and colimits
  12. Other colimits: coequalizers and pushouts
  13. Other limits: equalizers and pullbacks
  14. Prospects for future research

Key Result

Proposition 3.2

The initial object of $\mathsf{AtoMon}$ is the trivial monoid $\mathbb{0}:=\{1\}$, while its terminal object is the monoid $\mathbb{1}:=\{1, 0, a\}$ consisting of an identity $1$, an absorbing element $0$, and an element $a$ such that $a^2=0$. In particular, $\mathsf{AtoMon}$ does not have a zero ob

Theorems & Definitions (52)

  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Lemma 4.1
  • proof
  • Definition 4.2
  • ...and 42 more