  Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the
additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and
determine its generators (indecomposable integers) and relations; they can be
nicely described in terms of the periodic continued fraction for $\sqrt D$. We
also characterize all uniquely decomposable integers in $K$ and estimate their
norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$
completely determines the real quadratic field $K$.
