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 $(P,\leq _{P})$ 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:

\{x,y\}\subset P

, and

x\leq _{P}y

;

\{x,y\}\subset Q

, and

x\leq _{Q}y

; or

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.

References

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

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Star product

Definition

Example

Properties

See also

References