学术报告(Salvatore Tringali 12.10)

Additive decompositions of sets of integers into irreducible sets (对整数集的子集的不可约加法分解)

发布人:杨晓静 发布日期:2019-11-25
主题
Additive decompositions of sets of integers into irreducible sets (对整数集的子集的不可约加法分解)
活动时间
-
活动地址
数学学院 新数学楼415室
主讲人
Salvatore Tringali副教授 河北师范大学数学科学学院

时间:2019年12月10日 16:00-17:00

摘要:Let Z be the ring of integers. A subset A of Z is called irreducible if |A| ≠ 1 and AX + Y for all subsets X, Y of Z such that neither X nor Y is a singleton. Accordingly, if X is a subset of Z and |X| 2, we denote by L(X) the set of all integers n 1 for which there exist irreducible subsets A1, ... , An of Z such that

X=A1+...+An :={a1+...+an : a1A1, ... , an∈An已移除图像。}.

It was conjectured in [1, § 5] that, for every non-empty finite subset L of Z2, there exists a subset X of Z such that L(X) = L. I will survey what (little) is known about this conjecture and frame the problem within the broader context of factorization theory.

Reference:

[1] Y. Fan, S. Tringali, Power monoids: A bridge between factorization theory and arithmetic combinatorics, J. Algebra 512 (2018) 252-294.

Z是整数环。Z的一个子集A称为不可约的,若A的元素个数不为1,且A不能写成两个Z中集合XY之和,其中X和Y都不是独元集。因此,对一个至少有两个元素的Z的子集X,记L(X)为所有满足下面条件的正整数之集:X可以表为n个不可约集之和。

我们猜想,对每个非空集合L属于大于等于2的自然数集,都存在一个X使得L(X)=L。我们综述一下关于这个猜想的已知结果,以及把它放入一个更大的分解理论框架中。