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 , ≤ P ) {\displaystyle (P,\leq _{P})} and ( Q , ≤ Q ) {\displaystyle (Q,\leq _{Q})} , where P {\displaystyle P} has a unique maximal element 1 ^ {\displaystyle {\widehat {1}}} and Q {\displaystyle Q} has a unique minimal element 0 ^ {\displaystyle {\widehat {0}}} , is a poset P ∗ Q {\displaystyle P*Q} on the set ( P ∖ { 1 ^ } ) ∪ ( Q ∖ { 0 ^ } ) {\displaystyle (P\setminus \{{\widehat {1}}\})\cup (Q\setminus \{{\widehat {0}}\})} . We define the partial order ≤ P ∗ Q {\displaystyle \leq _{P*Q}} by x ≤ y {\displaystyle x\leq y} if and only if:
1. { x , y } ⊂ P {\displaystyle \{x,y\}\subset P} , and x ≤ P y {\displaystyle x\leq _{P}y} ; 2. { x , y } ⊂ Q {\displaystyle \{x,y\}\subset Q} , and x ≤ Q y {\displaystyle x\leq _{Q}y} ; or 3. x ∈ P {\displaystyle x\in P} and y ∈ Q {\displaystyle y\in Q} . In other words, we pluck out the top of P {\displaystyle P} and the bottom of Q {\displaystyle Q} , and require that everything in P {\displaystyle P} be smaller than everything in Q {\displaystyle Q} .
Example For example, suppose P {\displaystyle P} and Q {\displaystyle Q} are the Boolean algebra on two elements.
Then P ∗ Q {\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 {\displaystyle f} -vectors and the c d {\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.
