Green–Lazarsfeld property \(N_p\) for Segre product of Hibi rings

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