  Let $\Gamma=\langle \alpha, \beta \rangle$ be a numerical semigroup. In this
article we consider the dual $\Delta^*$ of a $\Gamma$-semimodule $\Delta$; in
particular we deduce a formula that expresses the minimal set of generators of
$\Delta^*$ in terms of the generators of $\Delta$. As applications we compute
the minimal graded free resolution of a graded
$\mathbb{F}[t^{\alpha},t^{\beta}]$-submodule of $\mathbb{F}[t]$, and we
investigate the structure of the selfdual $\Gamma$-semimodules, leading to a
new way of counting them.
