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 $P=Q=B_2$ $$\xymatrix{ & \widehat{1}_P\ar@{-}[dl]\ar@{-}[dr] & & & & \widehat{1}_Q\ar@{-}[dl]\ar@{-}[dr] & \\ a_P\ar@{-}[dr] & & b_P\ar@{-}[dl] & & a_Q\ar@{-}[dr] & & b_Q\ar@{-}[dl] \\ & \widehat{0}_P & & & & \widehat{0}_Q & }$$ Then $P*Q$ is the poset with the Hasse diagram below. $$\xymatrix{ & \widehat{1}_Q\ar@{-}[dl]\ar@{-}[dr] & \\ a_Q\ar@{-}[d]\ar@{-}[drr] & & b_Q\ar@{-}[dll]\ar@{-}[d] \\ a_P\ar@{-}[dr] & & b_P\ar@{-}[dl] \\ & \widehat{0}_P & }$$ The star product of Eulerian posets is Eulerian.

Bibliography

1

Stanley, R., Flag $f$ vectors and the $\mathbf{cd}$ index, Math. Z. 216 (1994), 483-499.