--- a/pnas/pnas.tex	Wed Mar 30 07:16:14 2011 -0700
+++ b/pnas/pnas.tex	Wed Mar 30 08:03:27 2011 -0700
@@ -743,7 +743,7 @@
 
 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
-with $\du_b Y_b = M_i$ for some $i$.
+with $\du_b Y_b = M_i$ for some $i$, and each $M_j$ with $j<i$ is also a disjoint union of balls.
 
 \begin{defn}
 The poset $\cell(W)$ has objects the permissible decompositions of $W$,