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.

DefinitionEdit

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  .

ExampleEdit

For example, suppose   and   are the Boolean algebra on two elements.

 

Then   is the poset with the Hasse diagram below.

 

PropertiesEdit

The star product of Eulerian posets is Eulerian.

See alsoEdit

ReferencesEdit

  • Stanley, R., Flag  -vectors and the  -index, Math. Z. 216 (1994), 483-499.


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