# 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.

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

## 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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.