Star product

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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.


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.


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

Star product 1.png

Then P*Q is the poset with the Hasse diagram below.

Star product 3.png


The star product of Eulerian posets is Eulerian.

See also[edit]


  • 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.