Given any cancellative continuous semigroup operation $$\star $$ ⋆ on the positive real numbers $$\mathbf {R}_+$$ R + with the ordinary topology, we completely characterize the set $$\mathcal {D}_\star (\mathbf {R}_+)$$ D ⋆ ( R + ) of all cancellative continuous semigroup operations on $$\mathbf {R}_+$$ R + which are distributed by $$\star $$ ⋆ in terms of homeomorphism. As a consequence, we show that an arbitrary semigroup operation in $$\mathcal {D}_\star (\mathbf {R}_+)$$ D ⋆ ( R + ) is homeomorphically isomorphic to the ordinary addition $$+$$ + on $$\mathbf {R}_+$$ R + .

中文翻译：

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正实数上的分配半群运算的刻画

给定任何对正实数 $$\mathbf {R}_+$$ R + 的取消连续半群运算 $$\star $$ ⋆ 具有普通拓扑，我们完全刻画了集合 $$\mathcal {D}_\ $$\mathbf {R}_+$$ R + 上的所有取消连续半群运算的star (\mathbf {R}_+)$$ D ⋆ ( R + ) 分布于$$\star $$ ⋆同胚的术语。因此，我们证明 $$\mathcal {D}_\star (\mathbf {R}_+)$$ D ⋆ ( R + ) 中的任意半群运算与普通加法 $$+$ 同构同构$ + 在 $$\mathbf {R}_+$$ R + 上。