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 ${\displaystyle (P,\leq _{P})}$  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 ${\displaystyle Q}$ .

Example

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.

Properties

The star product of Eulerian posets is Eulerian.

References

• Stanley, R., Flag ${\displaystyle f}$ -vectors and the ${\displaystyle \mathbf {cd} }$ -index, Math. Z. 216 (1994), 483-499.

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