A formula for the conductor of a semimodule of a numerical semigroup with two generators

We provide an expression for the conductor $c(\mathrm{\Delta })$ of a semimodule $\mathrm{\Delta }$ of a numerical semigroup $\mathrm{\Gamma }$ with two generators in terms of the syzygy module of $\mathrm{\Delta }$ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of $\mathrm{\Gamma }$, as well as a formula for $c(\mathrm{\Delta })$ in terms of the dual semimodule of $\mathrm{\Delta }$.