containing the rightmost embedding, that is, the embedding where the nonzero letters are the furthest right, of each equivalence class of $E^{\sigma,\pi}\slash\sim$.

Consider $\eta\in \widehat{E}^{\sigma,\pi}$ and define the zero set of $\eta$ as $Z(\eta)=\{i\,|\, \eta_i=0\}$. Define $EZ^{\sigma,\pi}$ to be the set of sets of embeddings in $\widehat{E}^{\sigma,\pi}$ such that for each set $S\in EZ^{\sigma,\pi}$ we have $\bigcap_{\eta\in S}Z(\eta)=\emptyset$.

When defining $\widehat{E}^{\sigma,\pi}$ we choose the rightmost embedding to ensure that all normal embeddings are in $\widehat{E}^{\sigma,\pi}$. Note that if $\eta\sim\psi$ then $\hat{\eta}=\hat{\psi}$, which can be used as an equivalent definition of the equivalence relation. The set $EZ^{\sigma,\pi}$ is upwards closed under containment because if we take any set $S\in EZ^{\sigma,\pi}$ adding a new embedding to $S$ will result in a set that still has empty intersection of zero sets.

\[ex:3\] If $\sigma=132$ and $\pi=413265$ then the embedding $013200$ has zero set $Z(013200)=\{1,5,6\}$ and $$\begin{aligned}
E^{\sigma,\pi}=&\{013200,\,400065,\,010065,\,003065,\,000265\},\\ \widehat{E}^{\sigma,\pi}=&\{013200,\,400065,\,010065,\,000265\}, \\EZ^{\sigma,\pi}=&\{\{013200,\,400065\},\{013200,\,400065,\,010065\},\\ &\{013200,\,400065,\,000265\},\{013200,\,400065,\,010065,\,000265\}\}.\end{aligned}$$

Using our decomposition we build posets from embeddings in the following way:

Given an embedding $\eta\in E^{\sigma,\pi}$ define the poset $P(\eta):=[\hat{\eta}_1,\hat{\pi}_1]\times\cdots\times[\hat{\eta}_t,\hat{\pi}_t]$ and $$A^{\sigma,\pi}:=\bigcup_{\eta\in \widehat{E}^{\sigma,\pi}}P(\eta)^o,$$ where $P(\eta)^o$ denotes the interior of $P(\eta)$, that is, $P(\eta)$ with the top and bottom elements removed.

Consider $[132,413265]$ and let $\eta_1,\eta_2,\eta_3$ and $\eta_4$ be the embeddings listed in 