Green-Lazarsfeld property $N_p$ for Segre product of Hibi rings

In this article, we prove that if a Hibi ring satisfies property $N_2$, then its Segre product with a polynomial ring in finitely many variables also satisfies property $N_2$. When the polynomial ring is in two variables, we also prove the above statement for $N_3$. Moreover, we study the minimal Koszul relations of the second syzygy module of Hibi rings.