Semistar operations on Dedekind domains

We give an explicit description of the lattice $\text{Semistar}(D)$ of all semistar operations on any Dedekind domain $D$ from its set $\text{Max}(D)$ of maximal ideals. This descpription is constructive if $\text{Max}(D)$ is finite. As a corollary we show that $2^{\left(\genfrac{}{}{0px}{}{n}{[n\mathrm{/}2]}\right)}\le \mathrm{\mid }\text{Semistar}(D)\mathrm{\mid }\le 2^{2^n}$ if $n = |\Max(D)|$isfinite;wecompute$|\Semistar(D)|$if$|\Max(D)| \leq 7$; and we show that if $\text{Max}(D)$ is infinite then $\text{Semistar}(D)$ has cardinality $2^{2^{\mathrm{\mid }\text{Max}(D)\mathrm{\mid }}}$.