Star 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 $$(P,\le_P)$$ 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.