Star product

In mathematics, 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 a poset $P*Q$ on the set $(Psetminuswidehat\{1\})cup(Qsetminuswidehat\{0\})$. We define the partial order $le\_\{P*Q\}$ by $xle\; y$ if and only if:

:1. $\{x,y\}subset\; P$, and $xle\_P\; y$;:2. $\{x,y\}subset\; Q$, and $xle\_Q\; y$; or:3. $xin\; P$ and $yin\; Q$.

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 $P$ and $Q$ are the Boolean algebra on two elements.

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

The star product of Eulerian posets is Eulerian.

Bibliography

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