Star product explained

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 cd -index, Math. Z. 216 (1994), 483-499.