$\widehat{E}^{\sigma,\pi}$ in Example \[ex:3\]. Then $\hat{\pi}=(1,1,21,21)$ and $\hat{\eta}_1=(\emptyset,1,21,\emptyset)$, $\hat{\eta}_2=(1,\emptyset,\emptyset,21)$, $\hat{\eta}_3=(\emptyset,1,\emptyset,21)$ and $\hat{\eta}_4=(\emptyset,\emptyset,1,21)$. See Figure \[fig:exA\] for $P(\eta_i)$ and $A^{132,413265}$.

The poset $A^{\sigma,\pi}$ consists of the elements $\hat{\eta}$ for all $\eta\in\widehat{E}^{\lambda,\pi}$ and all $\lambda\in(\sigma,\pi)$. Therefore, we define a surjective poset map $f$ from $A^{\sigma,\pi}$ to $(\sigma,\pi)$ in the following way:

\[defn:map\] Let $f:A^{\sigma,\pi}\rightarrow(\sigma,\pi)$ be the map which maps all elements $\hat{\eta}$, where $\eta\in\widehat{E}^{\lambda,\pi}$, to $\lambda$.

If $[132,413265]$ and $\hat{\eta}=(1,\emptyset,1,21)$ then $\eta=400265\in \widehat{E}^{2143,\pi}$, so $f(\hat{\eta})=2143$.

The Topology of a Poset
-----------------------

We study the topology of a poset by constructing a simplicial complex from the poset in the following way:

Let $P$ be a poset. A *chain* in $P$ is a totally ordered subset $\{z_1<\cdots<z_t\}$. The *order complex* of $P$, denoted $\Delta(P)$, is the simplicial complex whose vertices are the elements of $P$ and whose faces are the chains of $P$.

When we refer to the order complex of an interval $[\sigma,\pi]$ we mean the order complex of the interior $(\sigma,\pi)$, which we denote $\Delta(\sigma,\pi)$.

Consider the interval $I=[123,4567123]$. An example of a chain in $(123,4567123)$ is $4123<456123$. The order complex and Hasse diagram of $I$ are given in Figure \[fig:ordercomplex\].

We refer to a poset and its order complex interchangeably, so 