  Sch\"{u}tzenberger's promotion operator is an extensively-studied bijection
that permutes the linear extensions of a finite poset. We introduce a natural
extension $\partial$ of this operator that acts on all labelings of a poset. We
prove several properties of $\partial$; in particular, we show that for every
labeling $L$ of an $n$-element poset $P$, the labeling $\partial^{n-1}(L)$ is a
linear extension of $P$. Thus, we can view the dynamical system defined by
$\partial$ as a sorting procedure that sorts labelings into linear extensions.
For all $0\leq k\leq n-1$, we characterize the $n$-element posets $P$ that
admit labelings that require at least $n-k-1$ iterations of $\partial$ in order
to become linear extensions. The case in which $k=0$ concerns labelings that
require the maximum possible number of iterations in order to be sorted; we
call these labelings tangled. We explicitly enumerate tangled labelings for a
large class of posets that we call inflated rooted forest posets. For an
arbitrary finite poset, we show how to enumerate the sortable labelings, which
are the labelings $L$ such that $\partial(L)$ is a linear extension.
