Tensor products in $\beta({\mathbb N}\times{\mathbb N})$

Abstract: Given a discrete space $S$, the Stone-Čech compactification $\beta S$ of $S$ consists of all of the ultrafilters on $S$. If $p\in\beta S$ and $q\in\beta T$, then the {\it tensor product\/}, $p\otimes q\in \beta (S\times T)$ is defined by $$p\otimes q=\{A\subseteq S\times T:\{x\in S:\{y\in T:(x,y)\in A\}\in q\}\in p\}\,.$$ Tensor products of members of $\beta {\mathbb N}$ are intimately related to addition on ${\mathbb N}$. If $\sigma:{\mathbb N}\times{\mathbb N}\to{\mathbb N}$ is defined by $\sigma(s,t)=s+t$ and $\widetilde \sigma:\beta({\mathbb N}\times{\mathbb N})\to \beta {\mathbb N}$ is its continuous extension, then for any $p,q\in\beta{\mathbb N}$, $\widetilde\sigma(p\otimes q)=p+q$. Among our results are the facts that if $S=({\mathbb N},+)$ or $S=({\mathbb R}_d,+)$, where ${\mathbb R}_d$ is ${\mathbb R}$ with the discrete topology, and $S^*=\beta S\setminus S$, then $S^*\otimes S^*$ misses the closure of the smallest ideal of $\beta(S\times S)$ and $\beta S\otimes\beta S$ is not a Borel subset of $\beta(S\times S)$.

Joint work with Dona Strauss.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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