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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.

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