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
and
, where
has a unique maximal element
and
has a unique minimal element
, is a poset
on the set
. We define the partial order
by
if and only if:
- 1.
, and
;
- 2.
, and
; or
- 3.
and
.
In other words, we pluck out the top of
and the bottom of
, and require that everything in
be smaller than everything in
.
Example
For example, suppose
and
are the Boolean algebra on two elements.
Then
is the poset with the Hasse diagram below.
Properties
The star product of Eulerian posets is Eulerian.