论文标题
关于多项式半程的代数估值的添加结构
On the additive structure of algebraic valuations of polynomial semirings
论文作者
论文摘要
在本文中，我们研究了积极代数估值的添加剂单体$ \ mathbb {n} _0 [α] $多项式的半度性的$ \ mathbb {n} _0 [x] $，该方法使用每种方法。不可转化的元素因素成不可减数。我们首先要确定$ \ Mathbb {n} _0 [α] $是原子的，我们对其一组不可减数的描述进行了明确的描述。如果每个元素仅具有有限的因素化（按顺序和同伴），则原子单体是一个有限的分解单体（FFM），并且如果每个元素在其每个分数中都有不可估算的数量（计数重复）的数量，则它是一个有界的分解单体化（BFM）。我们表明，对于MONOID $ \ MATHBB {N} _0 [α] $，成为BFM的属性和成为FFM的属性等同于主理想的上升链条件（ACCP）。最后，我们给出了$ \ mathbb {n} _0 [α] $的各种特征，使其成为唯一的分解单体化（UFM），其中两个就$α$的最小多项式而言。在此过程中，还研究了有限生成，半因素和长度因素的特性。
In this paper, we study factorizations in the additive monoids of positive algebraic valuations $\mathbb{N}_0[α]$ of the semiring of polynomials $\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $\mathbb{N}_0[α]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid $\mathbb{N}_0[α]$, the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for $\mathbb{N}_0[α]$ to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of $α$. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.
