We will prove that, for any abelian group $G$, the canonical (surjective and continuous) mapping $\boldsymbol{\beta}G \to {\frak b}G$ from the Stone-\v{C}ech compactification $\boldsymbol{\beta}G$ of $G$ to its Bohr compactfication ${\frak b}G$ is a homomorphism with respect to the semigroup operation on $\boldsymbol{\beta}G$, extending the multiplication on $G$, and the group operation on ${\frak b}G$. Moreover, the Bohr compactification ${\frak b}G$ is canonically isomorphic (both in algebraic and topological sense) to the quotient of $\boldsymbol{\beta}G$ with respect to the least closed congruence relation on $\boldsymbol{\beta}G$ merging all the Schur ultrafilters on $G$ into the unit of $G$.

中文翻译：

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阿贝尔群的玻尔紧化作为其 Stone-Čech 紧化的商

我们将证明，对于任何阿贝尔群 $G$，正则（满射和连续）映射 $\boldsymbol{\beta}G \to {\frak b}G$ 从 Stone-\v{C}ech 紧化 $ $G$ 的\boldsymbol{\beta}G$ 到其玻尔紧缩${\frak b}G$ 是关于$\boldsymbol{\beta}G$ 上的半群运算的同态，扩展了$G 上的乘法$，以及 ${\frak b}G$ 上的群运算。此外，玻尔紧缩 ${\frak b}G$ 与 $\boldsymbol{\beta}G$ 关于 $\boldsymbol{ 上的最小同余关系的商是典型同构的（在代数和拓扑意义上） \beta}G$ 将 $G$ 上的所有 Schur 超滤器合并为 $G$ 的单位。