Pseudo-Frobenius numbers and defining ideals in stretched numerical semigroup rings

The pseudo-Frobenius numbers of a numerical semigroup $H$ are deeply connected to the structure of the defining ideal of its semigroup ring $k[H]$. In this paper, we resolve a certain conjecture related to this connection under the assumption that $k[H]/(t^a)$ is stretched, where $a$ is the multiplicity of $H$. Furthermore, we provide numerical conditions for the tangent cone of $k[H]$ to be Cohen-Macaulay.