Not to be confused with the Moyal product, also sometimes referred to as the 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.
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.
star, product, confused, with, moyal, product, also, sometimes, referred, star, product, mathematics, star, product, method, combining, graded, posets, with, unique, minimal, maximal, elements, preserving, property, that, posets, eulerian, contents, definition. Not to be confused with the Moyal product also sometimes referred to as the 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 Contents 1 Definition 2 Example 3 Properties 4 See also 5 ReferencesDefinition editThe star product of two graded posets P P displaystyle P leq P nbsp and Q Q displaystyle Q leq Q nbsp where P displaystyle P nbsp has a unique maximal element 1 displaystyle widehat 1 nbsp and Q displaystyle Q nbsp has a unique minimal element 0 displaystyle widehat 0 nbsp is a poset P Q displaystyle P Q nbsp on the set P 1 Q 0 displaystyle P setminus widehat 1 cup Q setminus widehat 0 nbsp We define the partial order P Q displaystyle leq P Q nbsp by x y displaystyle x leq y nbsp if and only if 1 x y P displaystyle x y subset P nbsp and x Py displaystyle x leq P y nbsp 2 x y Q displaystyle x y subset Q nbsp and x Qy displaystyle x leq Q y nbsp or 3 x P displaystyle x in P nbsp and y Q displaystyle y in Q nbsp In other words we pluck out the top of P displaystyle P nbsp and the bottom of Q displaystyle Q nbsp and require that everything in P displaystyle P nbsp be smaller than everything in Q displaystyle Q nbsp Example editFor example suppose P displaystyle P nbsp and Q displaystyle Q nbsp are the Boolean algebra on two elements nbsp Then P Q displaystyle P Q nbsp is the poset with the Hasse diagram below nbsp Properties editThe star product of Eulerian posets is Eulerian See also editProduct order a different way of combining posetsReferences editStanley R Flag f displaystyle f nbsp vectors and the cd displaystyle mathbf cd nbsp index Math Z 216 1994 483 499 This article incorporates material from star product on PlanetMath which is licensed under the Creative Commons Attribution Share Alike License Retrieved from https en wikipedia org w index php title Star product amp oldid 1127632167, wikipedia, wiki, book, books, library,