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Factorization in Finitely-Presented Monoids

Alfred Geroldinger, Zachary Mesyan

Abstract

We study arithmetic properties of factorizations of elements into products of generators, in monoids given with explicit presentations. After relating and comparing this perspective to the more usual approach of factoring into products of atoms, as well as other more recent alternatives, we explore how the relations in the presentation of a monoid affect factorization. In the process, we construct a large class of non-commutative fully elastic monoids. We also show that any finitely-presented cancellative normalizing monoid satisfies the Structure Theorem for Unions. Examples are constructed to demonstrate the sharpness of our results, and exhibit unusual factorization behavior.

Factorization in Finitely-Presented Monoids

Abstract

We study arithmetic properties of factorizations of elements into products of generators, in monoids given with explicit presentations. After relating and comparing this perspective to the more usual approach of factoring into products of atoms, as well as other more recent alternatives, we explore how the relations in the presentation of a monoid affect factorization. In the process, we construct a large class of non-commutative fully elastic monoids. We also show that any finitely-presented cancellative normalizing monoid satisfies the Structure Theorem for Unions. Examples are constructed to demonstrate the sharpness of our results, and exhibit unusual factorization behavior.

Paper Structure

This paper contains 11 sections, 20 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Terminology
  4. Subadditive Families
  5. Factorizations
  6. Structure Theorem for Unions
  7. Generators vs. Atoms and Quarks
  8. Basic Observations
  9. Normalizing Monoids
  10. Examples
  11. Summary

Key Result

Proposition 1

Let $M := \langle X \mid R \rangle$, where $X \neq \emptyset$.

Theorems & Definitions (41)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • Proposition 6
  • ...and 31 more