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 $2 = d(d(1,2),3) \neq d(1,d(2,3)) = 0$.)