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Is there an associative operation $\star \colon \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ wich also happens to be a metric on $\mathbb{R}_{\geq 0}$? Furthermore is there a monoid/group structure with such an operation? Would be the group topological in respect to the metric-induced topology?

(The usual metric given by $d(x,y) = \lvert y - x \rvert$ for $x, y \in \mathbb{R}_{\geq 0}$ is not associative since $2 = d(d(1,2),3) \neq d(1,d(2,3)) = 0$.)

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up vote 2 down vote accepted

Yes, it is true.

Please check this link http://mathoverflow.net/questions/16214/is-there-an-associative-metric-on-the-non-negative-reals .

Thanks.

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Oh nice. I couldn't find anything on this site, but I didn't look on mathoverflow. –  k.stm Oct 8 '12 at 9:25
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