star product

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 the 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 $Q$ . For example, suppose $P=Q=B_{2}$ .

\xymatrix{&\widehat{1}_{P}\ar@{-}[dl]\ar@{-}[dr]&&&&\widehat{1}_{Q}\ar@{-}[dl]% \ar@{-}[dr]&\\ a_{P}\ar@{-}[dr]&&b_{P}\ar@{-}[dl]&&a_{Q}\ar@{-}[dr]&&b_{Q}\ar@{-}[dl]\\ &\widehat{0}_{P}&&&&\widehat{0}_{Q}&}

Then $P*Q$ is the poset with the Hasse diagram below.

\xymatrix{&\widehat{1}_{Q}\ar@{-}[dl]\ar@{-}[dr]&\\ a_{Q}\ar@{-}[d]\ar@{-}[drr]&&b_{Q}\ar@{-}[dll]\ar@{-}[d]\\ a_{P}\ar@{-}[dr]&&b_{P}\ar@{-}[dl]\\ &\widehat{0}_{P}&}

The star product of Eulerian posets is Eulerian.

References

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

Title	star product
Canonical name	StarProduct
Date of creation	2013-03-22 14:09:17
Last modified on	2013-03-22 14:09:17
Owner	mps (409)
Last modified by	mps (409)
Numerical id	4
Author	mps (409)
Entry type	Definition
Classification	msc 05E99
Classification	msc 06A11
Related topic	Poset
Related topic	GradedPoset