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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.)
##### Keywords:
semigroup; Stone-Čech compactification; circle group
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