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Power monoids and their arithmetic: a survey

Salvatore Tringali

Abstract

The non-empty finite subsets of a multiplicatively written monoid form a monoid in their own right, and so do the finite subsets that contain the identity element. Partly due to their unusual arithmetic properties, these structures, known as power monoids, have attracted increasing attention in recent years and have in turn stimulated growing interest in new perspectives in factorization theory, better suited to non-cancellative settings. We survey these developments and briefly review some related aspects.

Power monoids and their arithmetic: a survey

Abstract

The non-empty finite subsets of a multiplicatively written monoid form a monoid in their own right, and so do the finite subsets that contain the identity element. Partly due to their unusual arithmetic properties, these structures, known as power monoids, have attracted increasing attention in recent years and have in turn stimulated growing interest in new perspectives in factorization theory, better suited to non-cancellative settings. We survey these developments and briefly review some related aspects.
Paper Structure (15 sections, 20 theorems, 24 equations)

This paper contains 15 sections, 20 theorems, 24 equations.

Table of Contents

  1. Introduction.
  2. Isomorphism problems.
  3. Power monoids.
  4. Symmetry and rigidity.
  5. Formalities.
  6. New perspectives on an old classic.
  7. Irreducibles vs atoms.
  8. Factorizations and lengths.
  9. Arithmetic in power monoids.
  10. Overture.
  11. Interlude on cancellation.
  12. Minuetto of irreducibles.
  13. Finale with lengths.
  14. Into the future.
  15. Acknowledgments.

Key Result

Proposition 3.1

Every atom of a monoid is a quark.

Theorems & Definitions (41)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • ...and 31 more