# 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 and , where has a unique maximal element and has a unique minimal element , is a poset on the set . We define the partial order by if and only if:

- 1. , and ;
- 2. , and ; or
- 3. and .

In other words, we pluck out the top of and the bottom of , and require that everything in be smaller than everything in .

## Example

For example, suppose and are the Boolean algebra on two elements.

Then 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 -vectors and the -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.*