Separating linear expressions in the Stone-Čech compactification of direct sums

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• 作者：Neil Hindman^a ; ¹ ; ^{nhindman@aol.com" class="auth_mail" title="E-mail the corresponding author} ; ; Dona Strauss^b ; ^{d.strauss@hull.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：54D35 ; 22A15 ; 05D10
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 November 2016
• 年：2016
• 卷：213
• 期：Complete
• 页码：199-211
• 全文大小：400 K

文摘

A finite sequence $\overrightarrow{a}={〈a_i〉}_{i=1}^m$ in Z∖{0}$\mathbb{Z}\setminus \{0\}$ is compressed   provided a_i≠a_i+1$a_i\ne a_{i+1}$ for i<m$i<m$. It is known that if $\overrightarrow{a}={〈a_i〉}_{i=1}^m$ and $\overrightarrow{b}={〈b_i〉}_{i=1}^k$ are compressed sequences in Z∖{0}$\mathbb{Z}\setminus \{0\}$, then there exist idempotents p and q   in βQ_d∖{0}$\mathrm{<font\; color="red">\beta </font>}{\mathbb{Q}}_d\setminus \{0\}$ such that a₁p+a₂p+…+a_mp=b₁q+b₂q+…+b_kq$a_{1}p+a_{2}p+\dots +a_{m}p=b_{1}q+b_{2}q+\dots +b_{k}q$ if and only if $\overrightarrow{b}$ is a rational multiple of $\overrightarrow{a}$. In fact, if $\overrightarrow{b}$ is not a rational multiple of $\overrightarrow{a}$, then there is a partition of Q∖{0}$\mathbb{Q}\setminus \{0\}$ into two cells, neither of which is a member of a₁p+a₂p+…+a_mp$a_{1}p+a_{2}p+\dots +a_{m}p$ and a member of b₁q+b₂q+…+b_kq$b_{1}q+b_{2}q+\dots +b_{k}q$ for any idempotents p and q   in βQ_d∖{0}$\mathrm{<font\; color="red">\beta </font>}{\mathbb{Q}}_d\setminus \{0\}$. (Here βQ_d$\mathrm{<font\; color="red">\beta </font>}{\mathbb{Q}}_d$ is the Stone–Čech compactification of the set of rational numbers with the discrete topology.)

In this paper we extend these results to direct sums of Q$\mathbb{Q}$. As a corollary, we show that if $\overrightarrow{b}$ is not a rational multiple of $\overrightarrow{a}$ and G is any torsion free commutative group, then there do not exist idempotents p and q   in βG_d∖{0}$\mathrm{<font\; color="red">\beta </font>}{G}_d\setminus \{0\}$ such that a₁p+a₂p+…+a_mp=b₁q+b₂q+…+b_kq$a_{1}p+a_{2}p+\dots +a_{m}p=b_{1}q+b_{2}q+\dots +b_{k}q$. We also show that for direct sums of finitely many copies of Q$\mathbb{Q}$ we can separate the corresponding Milliken–Taylor systems, with a similar but weaker result for the direct sum of countably many copies of Q$\mathbb{Q}$.