  Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of
P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of
C(P) over a field K and characterize the level (resp. anticanonical level)
property of K[C(P)] by a combinatorial structure of P. In particular, we show
that if K[C(P)] is level (resp. anticanonical level), then so is K[O(P)]. We
exhibit examples which show the converse does not hold.
  Moreover, we show that the symbolic powers of the canonical ideal of K[C(P)]
are identical with ordinary ones and degrees of the generators of the canonical
and anticanonical ideals are consecutive integers.
