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.

The star product of two graded posets
(
P
,
≤
P
)
and
(
Q
,
≤
Q
)
, where
P
has a unique maximal element
1
^
and
Q
has a unique minimal element
0
^
, is a poset
P
∗
Q
on the set
(
P
∖
{
1
^
}
)
∪
(
Q
∖
{
0
^
}
)
. We define the partial order
≤
P
∗
Q
by
x
≤
y
if and only if:

1.

{
x
,
y
}
⊂
P
, and

x
≤
P
y
;
2.

{
x
,
y
}
⊂
Q
, and

x
≤
Q
y
; or
3.

x
∈
P
and

y
∈
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.

Then
P
∗
Q
is the poset with the Hasse diagram below.

The star product of Eulerian posets is Eulerian.