Factorization under Local Finiteness Conditions

Abstract

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called $⪯$-factorizations, for the $⪯$-non-units of a (multiplicatively written) monoid $H$ endowed with a preorder $⪯$, where an element $u\in H$ is a $⪯$-unit if $u⪯1_H⪯u$ and a $⪯$-non-unit otherwise. The ``building blocks'' of these factorizations are the $⪯$-irreducibles of $H$ (i.e., the $⪯$-non-units $a\in H$ that cannot be written as a product of two $⪯$-non-units each of which is strictly $⪯$-smaller than $a$); and it is interesting to look for sufficient conditions for the $⪯$-factorizations of a $⪯$-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal $⪯$-factorizations (i.e., a refinement of $⪯$-factorizations used to counter the ``blow-up phenomena'' that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the ``local behaviour'' of the pair $(H,⪯)$. Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where $⪯$ is the divisibility preorder on $H$ (and hence in the setup of the classical theory).

arXiv comment27 pages, 2 figures. Final version to appear in Journal of Algebra

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Cited by in corpus (8)

• A conjecture by Bienvenu and Geroldinger on power monoids
• On the atomic structure of torsion-free monoids
• On the finiteness of certain factorization invariants
• On power monoids and their automorphisms
• On transfer homomorphisms in commutative rings with zero-divisors
• On the arithmetic of power monoids
• Torsion groups and the Bienvenu--Geroldinger conjecture
• Factorization in Finitely-Presented Monoids