论文标题
无限阿贝尔组的通用零和加权零和加权零和加权
The universal zero-sum invariant and weighted zero-sum for infinite abelian groups
论文作者
论文摘要
令$ g $为一个Abelian集团，让$ \ Mathcal f（g）$为基础$ g $的免费交换性单体。对于$ω\ subset \ mathcal f（g）$，定义通用零和不变$ {\ Mathsf d}_Ω（g）$作为最小的整数$ \ ell $，以使每个序列$ t $ t $ to $ g $ g $ f length $ \ ell $都具有$ω$的subsequence。不变的$ {\ Mathsf D}_Ω（G）$统一了许多经典的零和不变性。令$ \ mathcal b（g）$为$ \ Mathcal f（g）$的submonoid，由$ g $上的所有零和序列组成，让$ \ mathcal a（g）$是由$ g $上的所有最小零sum序列组成的集合。在本文中，我们表明，除了几个特殊类别的组外，总是存在适当的子集$ \ $ \ mathcal a（g）$的适当子集$ω$，使得$ {\ mathsf d}_Ω（g）= {\ rm d}（g）$。此外，在有限循环基团的设置中，我们通过确定它们的交叉点讨论了所有最小集合的分布。 通过将通用的零和不变性与权重联系，我们在{\ sl sl infinite} abelian群体的情况下研究了零和问题。通用零和不变$ {\ Mathsf d} _ {ω; ψ}（g）$具有权重设置的$ψ$组的同构组的$ψ$均针对所有阿伯利亚组引入。加权的davenport常数$ {\ rm d}_ψ（g）$（作为权重的通用不变的一种特殊形式）也针对无限的阿贝尔组进行了研究。除其他结果外，我们还获得了必要和充分的条件，以便在$ |ψ| $是有限时，就权重$ {\ rm d}_ψ（g）<\ infty $而言。在这样做的过程中，通过使用Neumann定理，我们建立了通过cosets的某些给定子组的$ g $的有限封面与加权Davenport常数的有限范围之间的联系。
Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$. For $Ω\subset \mathcal F (G)$, define the universal zero-sum invariant ${\mathsf d}_Ω(G)$ to be the smallest integer $\ell$ such that every sequence $T$ over $G$ of length $\ell$ has a subsequence in $Ω$. The invariant ${\mathsf d}_Ω(G)$ unifies many classical zero-sum invariants. Let $\mathcal B (G)$ be the submonoid of $\mathcal F (G)$ consisting of all zero-sum sequences over $G$, and let $\mathcal A (G)$ be the set consisting of all minimal zero-sum sequences over $G$. In this paper, we show that except for a few special classes of groups, there always exists a proper subset $Ω$ of $\mathcal A (G)$ such that ${\mathsf d}_Ω(G)={\rm D}(G)$. Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of {\sl infinite} abelian groups. The universal zero-sum invariant ${\mathsf d}_{Ω; Ψ}(G)$ with weights set $Ψ$ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant ${\rm D}_Ψ(G)$ (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that ${\rm D}_Ψ(G)<\infty$ in terms of the weights set $Ψ$ when $|Ψ|$ is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group $G$ by cosets of some given subgroups of $G$, and the finiteness of weighted Davenport constant.
