Star product

The star product of two graded posets ${\displaystyle (P,\leq _{P})}$ and ${\displaystyle (Q,\leq _{Q})}$, where ${\displaystyle P}$ has a unique maximal element ${\displaystyle {\widehat {1}}}$ and ${\displaystyle Q}$ 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 ${\displaystyle x\leq y}$ if and only if:

1. ${\displaystyle \{x,y\}\subset P}$, and ${\displaystyle x\leq _{P}y}$;

2. ${\displaystyle \{x,y\}\subset Q}$, and ${\displaystyle x\leq _{Q}y}$; or

3. ${\displaystyle x\in P}$ and ${\displaystyle y\in Q}$.

In other words, we pluck out the top of ${\displaystyle P}$ and the bottom of ${\displaystyle Q}$, and require that everything in ${\displaystyle P}$ be smaller than everything in ${\displaystyle Q}$.

For example, suppose ${\displaystyle P}$ and ${\displaystyle Q}$ are the Boolean algebra on two elements.

Then ${\displaystyle P*Q}$ is the poset with the Hasse diagram below.