741 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls

   741 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls

   742 are glued up to yield $W$, and just require that there is some non-pathological way to do this. 

   742 are glued up to yield $W$, and just require that there is some non-pathological way to do this. 

   743 

   743 

   744 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement

   744 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement

   745 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$

   745 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$

   747 

   747 

   748 \begin{defn}

   748 \begin{defn}

   749 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 

   749 The poset $\cell(W)$ has objects the permissible decompositions of $W$, 

   750 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

   750 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

   751 See Figure \ref{partofJfig} for an example.

   751 See Figure \ref{partofJfig} for an example.