star product

The star product of two graded posets $(P,{\le }_P)$ and $(Q,{\le }_Q)$, where $P$ has a unique maximal element $\widehat{1}$ and $Q$ has a unique minimal element $\widehat{0}$, is the poset $P*Q$ on the set $(P\setminus \widehat{1})\cup (Q\setminus \widehat{0})$. We define the partial order ${\le }_{P*Q}$ by $x\le y$ if and only if:

1. $\{x,y\}\subset P$, and $x{\le }_{P}y$;

2. $\{x,y\}\subset Q$, and $x{\le }_{Q}y$; or

3. $x\in P$ and $y\in Q$.

In other words, we pluck out the top of $P$ and the bottom of $Q$, and require that everything in $P$ be smaller than everything in $Q$.For example, suppose $P=Q=B_2$.

Then $P*Q$ is the poset with the Hasse diagram below.

The star product of Eulerian posets is Eulerian.