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SUMMARY:Neil Hindman (Howard University)
DTSTART:20200604T180000Z
DTEND:20200604T182500Z
DTSTAMP:20260423T011321Z
UID:CANT2020/51
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/51/
 ">Tensor products in $\\beta({\\mathbb N}\\times{\\mathbb N})$</a>\nby Nei
 l Hindman (Howard University) as part of Combinatorial and additive number
  theory (CANT 2021)\n\n\nAbstract\nGiven a discrete space $S$\, the \nSton
 e-Čech compactification $\\beta S$ of $S$\nconsists of all of the ultrafi
 lters on $S$. If\n$p\\in\\beta S$ and $q\\in\\beta T$\, then the {\\it ten
 sor\nproduct\\/}\, $p\\otimes q\\in \\beta (S\\times T)$\nis defined by\n$
 $p\\otimes q=\\{A\\subseteq S\\times T:\\{x\\in S:\\{y\\in T:(x\,y)\\in A\
 \}\\in q\\}\\in p\\}\\\,.$$\nTensor products of members of $\\beta {\\math
 bb N}$ are intimately related to \naddition on ${\\mathbb N}$. If $\\sigma
 :{\\mathbb N}\\times{\\mathbb N}\\to{\\mathbb N}$ is\ndefined by $\\sigma(
 s\,t)=s+t$ and $\\widetilde \\sigma:\\beta({\\mathbb N}\\times{\\mathbb N}
 )\\to\n\\beta {\\mathbb N}$ is its continuous extension\, then for any $p\
 ,q\\in\\beta{\\mathbb N}$\,\n$\\widetilde\\sigma(p\\otimes q)=p+q$. Among 
 our results are the \nfacts that if $S=({\\mathbb N}\,+)$ or $S=({\\mathbb
  R}_d\,+)$\, where\n${\\mathbb R}_d$ is ${\\mathbb R}$ with the discrete t
 opology\, and $S^*=\\beta S\\setminus S$\, then\n$S^*\\otimes S^*$ misses 
 the closure of the smallest ideal of $\\beta(S\\times S)$ and\n$\\beta S\\
 otimes\\beta S$ is not a Borel subset of $\\beta(S\\times S)$. \n\nJoint w
 ork with  Dona Strauss.\n
LOCATION:https://researchseminars.org/talk/CANT2020/51/
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