  For any finite poset P, there is a natural operator $X$ acting on the
antichains of P. We discuss conjectural properties of this operator for some
graded posets associated with irreducible root systems. In particular, if
$\Delta^+$ is the set of positive roots and $\Pi$ is the set of simple roots in
$\Delta^+$, then we consider the cases $P=\Delta^+$ and $\Delta^+\setminus
\Pi$. For the root system of type $A_n$, we consider an $X$-invariant
integer-valued function on the set of antichains of $\Delta^+$ and establish
some properties of it.
