# 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 $(P,\leq _{P})$  and $(Q,\leq _{Q})$ , 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 $\leq _{P*Q}$  by $x\leq y$  if and only if:

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