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Linear equations in the Stone-Čech compactification of \(\mathbb N\). (English) Zbl 0953.22002
Let \(S\) be a discrete semigroup, and \(\beta S\) its Stone-Čech compactification. The semigroup operation on \(S\) may be extended to \(\beta S\) and in this way \(\beta S\) becomes a compact right topological semigroup (the mapping \(x\mapsto xa\) from \(\beta S\) into itself is continuous for each \(a\in\beta S\)). The authors show that the equations (i) \(u+a\cdot p=v+b\cdot p\) and (ii) \(a\cdot p+u=b\cdot p+v\) have no solutions for distinct positive integers \(a,b\) with \(u,v\in\beta {\mathbb N}\) and \(p\in \beta{\mathbb N}\setminus{\mathbb N}\). Equation (i) is shown to hold more generally for a large class of abelian groups, and this class is completely characterized. Equation (ii) is shown to hold for semigroups embeddable in the circle group.

MSC:
22A15 Structure of topological semigroups
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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