# star product

The star product of two graded posets $(P,\leq_{P})$ and $(Q,\leq_{Q})$, 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 $\leq_{P*Q}$ by $x\leq y$ if and only if:

1. 1.

$\{x,y\}\subset P$, and $x\leq_{P}y$;

2. 2.

$\{x,y\}\subset Q$, and $x\leq_{Q}y$; or

3. 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.

## References

• 1 Stanley, R., Flag $f$-vectors and the $\mathbf{cd}$-index, Math. Z. 216 (1994), 483-499.
Title star product StarProduct 2013-03-22 14:09:17 2013-03-22 14:09:17 mps (409) mps (409) 4 mps (409) Definition msc 05E99 msc 06A11 Poset GradedPoset