  In this article, we prove necessary conditions for Hibi rings to satisfy
Green-Lazarsfeld property $N_p$ for $p=2$ and $3$. We also show that if a Hibi
ring satisfies property $N_4$, then it is a polynomial ring or it has a linear
resolution. Therefore, it satisfies property $N_p$ for all $p\geq 4$ as well.
As a consequence, we characterize distributive lattices whose comparability
graph is chordal in terms of the subposet of join-irreducibles of the
distributive lattice. Moreover, we characterize complete intersection Hibi
rings.
