Star product

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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 the 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.