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 $\preceq$factorizations, for the $\preceq$nonunits of a (multiplicatively written) monoid $H$ endowed with a preorder $\preceq$, where an element $u \in H$ is a $\preceq$unit if $u \preceq 1_H \preceq u$ and a $\preceq$nonunit otherwise. The ``building blocks'' of these factorizations are the $\preceq$irreducibles of $H$ (i.e., the $\preceq$nonunits $a \in H$ that cannot be written as a product of two $\preceq$nonunits each of which is strictly $\preceq$smaller than $a$); and it is interesting to look for sufficient conditions for the $\preceq$factorizations of a $\preceq$nonunit 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 $\preceq$factorizations (i.e., a refinement of $\preceq$factorizations used to counter the ``blowup phenomena'' that are inherent to factorization in noncommutative or noncancellative monoids) and some finiteness conditions describing the ``local behaviour'' of the pair $(H, \preceq)$. 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 $\preceq$ is the divisibility preorder on $H$ (and hence in the setup of the classical theory).
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.05869
 arXiv:
 arXiv:2208.05869
 Bibcode:
 2022arXiv220805869C
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Commutative Algebra;
 Primary 20M10;
 20M13. Secondary 13A05;
 16U30;
 20M14
 EPrint:
 27 pages, 2 figures. Final version to appear in Journal of Algebra