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Some applications of a new approach to factorization

Laura Cossu

Abstract

As highlighted in a series of recent papers by Tringali and the author, fundamental aspects of the classical theory of factorization can be significantly generalized by blending the languages of monoids and preorders. Specifically, the definition of a suitable preorder on a monoid allows for the exploration of decompositions of its elements into (more or less) arbitrary factors. We provide an overview of the principal existence theorems in this new theoretical framework. Furthermore, we showcase additional applications beyond classical factorization, emphasizing its generality. In particular, we recover and refine a classical result by Howie on idempotent factorizations in the full transformation monoid of a finite set.

Some applications of a new approach to factorization

Abstract

As highlighted in a series of recent papers by Tringali and the author, fundamental aspects of the classical theory of factorization can be significantly generalized by blending the languages of monoids and preorders. Specifically, the definition of a suitable preorder on a monoid allows for the exploration of decompositions of its elements into (more or less) arbitrary factors. We provide an overview of the principal existence theorems in this new theoretical framework. Furthermore, we showcase additional applications beyond classical factorization, emphasizing its generality. In particular, we recover and refine a classical result by Howie on idempotent factorizations in the full transformation monoid of a finite set.
Paper Structure (13 sections, 19 theorems, 11 equations)

This paper contains 13 sections, 19 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Premonoids and abstract factorization theorems
  3. Premonoids, quarks, and irreducibles.
  4. Existence factorization theorems
  5. A review of the known applications
  6. Classical theorems
  7. Direct-sum decompositions of modules
  8. The Lasker-Noether theorem
  9. Power monoids
  10. Idempotent factorization of matrices
  11. Idempotent factorizations in r.fix-premonoids
  12. The r.fix-premonoid of the full transformation monoid
  13. Invertible linear maps over a finite dimensional vector space

Key Result

Theorem 2.2

Every Artinian premonoid is $s$-factorable for every integer $s\ge 2$ (and hence factorable).

Theorems & Definitions (38)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Proposition 2.5: Cos-Tri-2023(a)
  • Corollary 3.1
  • proof
  • Corollary 3.2: Tr20(c)
  • Corollary 3.3: Tr20(c)
  • ...and 28 more