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 \( {\displaystyle (P,\leq _{P})} \) and \( {\displaystyle (Q,\leq _{Q})} \), where P has a unique maximal element 1\( {\displaystyle {\widehat {1}}} \) and Q has a unique minimal element \( {\displaystyle {\widehat {0}}} \), is a poset \( {\displaystyle P*Q} \) on the set \( {\displaystyle (P\setminus \{{\widehat {1}}\})\cup (Q\setminus \{{\widehat {0}}\})} \). We define the partial order \( {\displaystyle \leq _{P*Q}} \) by \( x\leq y \) if and only if:
1. \( {\displaystyle \{x,y\}\subset P} \) , and x ≤ \( {\displaystyle x\leq _{P}y} \) ;
2. \( {\displaystyle \{x,y\}\subset Q} \) , and x ≤ \( {\displaystyle x\leq _{Q}y} \) ; or
3. \( {\displaystyle x\in P} \) and \( {\displaystyle y\in Q} \) .
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 \( {\displaystyle P*Q} \) 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 \) {\displaystyle \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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
