On the linearity of the syzygies of Hibi rings

In this paper, 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 Np for all p = 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.