In mathematics, the star product is a method of combining graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian.

## Definition

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 a 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. $\{x,y\}\subset P$ , and $x\leq _{P}y$ ;
2. $\{x,y\}\subset Q$ , and $x\leq _{Q}y$ ; 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 $Q$ .

## Example

For example, suppose $P$ and $Q$ are the Boolean algebra on two elements. Then $P*Q$ is the poset with the Hasse diagram below. ## Properties

The star product of Eulerian posets is Eulerian.

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