论文标题
在局部有限条件下进行分解
Factorization under Local Finiteness Conditions
论文作者
论文摘要
最近已经观察到，通过结合单体和预订的语言，可以极大地推广经典分解理论的基本方面。 This has led to various theorems on the existence of certain factorizations, herein called $\preceq$-factorizations, for the $\preceq$-non-units 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 $和$ \ prepeq $ -non-unit否则。这些因素化的``构建块''是$ h $ $ h $的$ \ preceq $ -Rreducibles（即，$ \ preceq $ -non-units $ a \ in H $ in H $无法写成两种$ \ preceq $ -non-units的产物，每个$ \ preceq $ -non-units cypec $ \ preceq $ \ preceq $ a $ a $ a $ a $ a $ a $ a $ a $有趣的是，寻找$ \ prepeq $ - factorization的足够条件，$ \ preceq $ -non-unit的长度或有限数（如果以合适的方式进行测量或计数）。这正是当前工作中解决的问题，其主要新颖性是对最小$ \ \ prepeq $ - 触发作用之间的相互作用的研究（即，$ \ prepeq $ - factorization的细化是对“ blig-up cormentive and“ blog-conmenation and contressive and consection and consection and consection”的范围，又是不合格的情况下的范围，是对未构成的范围的范围，并且是对某些范围的范围，并且是对不可测量的依据。 ``一对$（H，\ prepeq）$的``本地行为''。除了许多示例和备注之外，本文还包括许多算术结果，其中一部分在基本情况下已经是$ h $ $ h $的可划分的基本情况（因此在经典理论的设置中）。
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 $\preceq$-non-units 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 $\preceq$-non-unit otherwise. The ``building blocks'' of these factorizations are the $\preceq$-irreducibles of $H$ (i.e., the $\preceq$-non-units $a \in H$ that cannot be written as a product of two $\preceq$-non-units each of which is strictly $\preceq$-smaller than $a$); and it is interesting to look for sufficient conditions for the $\preceq$-factorizations of a $\preceq$-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 $\preceq$-factorizations (i.e., a refinement of $\preceq$-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 $(H, \preceq)$. 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).
