Duality and syzygies for semimodules over numerical semigroups

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.