幺半群代数中的极大公因子
Maximal common divisors in monoid algebras
- MIT（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究幺半群代数的极大公因子性质，证明预施赖埃尔幺半群的MCD性质可提升至域上幺半群代数，但原子性和q-GCD性质不提升，并识别新的秩-1无挠MCD及MCD-有限幺半群类。
AI中文摘要：
我们称一个交换幺半群 $M$ 具有 MCD 性质，如果 $M$ 的每个非空有限子集都有一个极大公因子（MCD）；并称 $M$ 具有 MCD-有限性质，如果 $M$ 的每个非空有限子集在相伴意义下仅有有限多个 MCD。众所周知，每个满足主理想升链条件的幺半群都是 MCD 幺半群，而每个有限分解幺半群和每个预施赖埃尔幺半群都是 MCD-有限幺半群。在本文中，我们在幺半群代数的框架下研究 MCD 和 MCD-有限性质。在识别出一类新的秩-$1$ 无挠 MCD 幺半群之后，我们研究了 MCD 性质到域上幺半群代数的提升，证明了如果一个预施赖埃尔幺半群具有 MCD 性质，那么它在任何域上的幺半群代数也具有 MCD 性质。然后我们证明，与多项式扩张的情形不同，当限制在 MCD 幺半群类时，原子性这一性质不会提升到域上的幺半群代数。在论文的第二部分，我们首先识别出一类新的秩-$1$ 无挠 MCD-有限幺半群。然后我们建立了 MCD-有限性质到多项式扩张的提升。我们以证明 q-GCD 性质（即每个非空有限子集至多有一个 MCD 的条件），这是比 MCD-有限性质更强的条件，即使限制在秩-$1$ 无挠幺半群类时，也不会提升到域上的幺半群代数来结束本文。
英文摘要：
We say that a commutative monoid $M$ has the MCD property if every nonempty finite subset of $M$ has a maximal common divisor (MCD), and we say that $M$ has the MCD-finite property if every nonempty finite subset of $M$ has only finitely many MCDs up to associates. It is well known that every monoid that satisfies the ascending chain condition on principal ideals is an MCD monoid, while every finite factorization monoid and every pre-Schreier monoid is an MCD-finite monoid. In this paper, we study the MCD and MCD-finite properties in the setting of monoid algebras. After identifying a new class of rank-$1$ torsion-free MCD monoids, we investigate the ascent of the MCD property to monoid algebras over fields, proving that if a pre-Schreier monoid has the MCD property then its monoid algebras over any field also have the MCD property. Then we prove that, unlike for the case of polynomial extensions, the property of being atomic does not ascend to monoid algebras over fields when restricted to the class of MCD monoids. In the second part of the paper, we first identify a new class of rank-$1$ torsion-free MCD-finite monoids. Then we establish the ascent of the MCD-finite property to polynomial extensions. We conclude the paper proving that the q-GCD property (i.e., the condition that every nonempty finite subset has at most one MCD), which is a condition stronger than the MCD-finite property, does not ascend to monoid algebras over fields even when restricted to the class of rank-$1$ torsion-free monoids.
