Star product

Not to be confused with the Moyal product, also sometimes referred to as the star 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 $(P,\leq _{P})$ 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.

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.