关于多项式半环代数赋值的加法结构
On the additive structure of algebraic valuations of polynomial semirings
摘要：
本文利用D。D.安德森，D。F. Anderson和M. 1990年的Zafrullah。一个可消交换么半群是原子幺半群，如果每个不可逆元素都分解为不可约元素。我们首先确定当$\mathbb{N}_0[α]$是原子的，我们给出了一个明确的描述，其不可约集。一个原子幺半群是一个有限分解幺半群（FFM），如果每个元素只有1000个分解（直到阶和相关），并且它是一个有界分解幺半群（BFM），如果对于每个元素，每个分解中的不可约数（计数重复）都有一个界。证明了对幺半群$\mathbb{N}_0[α]$，BFM的性质和FFM的性质等价于主理想上的升链条件条件（ACCP）.最后，我们给出了$\mathbb{N}_0[α]$是唯一分解么半群（UFM）的各种刻画，其中两个刻画是关于$α$的极小多项式的。同时，我们也研究了它的非线性生成、半阶乘和长度阶乘的性质。
英文摘要：
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.
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