  In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein
and almost Gorenstein and to be non-level and almost Gorenstein in terms of the
structure of the partially ordered set defining the Hibi ring. We also state a
criterion of a ladder determinantal ring defined by 2-minors to be
non-Gorenstein and almost Gorenstein in terms of the shape of the ladder.
