Star product

The term "Star product" may also refer to the Moyal product.

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 $(P,\le_P)$ 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 $(P\setminus\{\widehat{1}\})\cup(Q\setminus\{\widehat{0}\})$ . 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 $Q$ .

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.

See also

Product order, a different way of combining posets

References

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

This article incorporates material from star product on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.