Pseudo-Gorenstein and level Hibi rings

We introduce pseudo-Gorenstein rings and characterize those Hibi rings attached to a finite distributive lattice L which are pseudo-Gorenstein. The characterization is given in terms of the poset of join-irreducible elements of L. We also present a necessary condition for Hibi rings to be level. Special attention is given to simple planar and hyper-planar lattices. Finally the pseudo-Gorenstein and level property of Hibi rings and generalized Hibi rings is compared with each other.