  For a reductive Lie algebra $\mathfrak h$ and a simple finite-dimensional
$\mathfrak h$-module $V$, the set of weights of $V$, $P(V)$, has a natural
poset structure. We consider antichains in the weight poset $P(V)$ and a
certain operator $\mathfrak X$ acting on antichains. Eventually, we impose
stronger constraints on $(\mathfrak h,V)$ and stick to the case in which
$\mathfrak h$ and $V$ are associated with a $Z$-grading of a simple Lie algebra
$\mathfrak g$. Then $V$ is a weight multiplicity free $\mathfrak h$-module and
$P(V)$ can be regarded as a subposet of $\Delta^+$, where $\Delta$ is the root
system of $\mathfrak g$. Our goal is to demonstrate that antichains in the
weight posets associated with $Z$-gradings of $\mathfrak g$ exhibit many good
properties similar to those of $\Delta^+$ that are observed earlier in arXiv:
math.CO 0711.3353 (=Ref. [14] in the text).
