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Math Help - anyone help me about incidence algebra?

  1. #1
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    anyone help me about incidence algebra?

    what is incidence algebra? what does it include? what is its applications? anyone send me lecture notes?
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  2. #2
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    Quote Originally Posted by bogazichili View Post

    what is incidence algebra?
    let R be a commutative domain and (I, \leq) a (not necessarily finite) partially ordered set with this property that for any i,j \in I with i \leq j, the set: [i,j]=\{k \in I: \ i \leq k \leq j \} is finite.

    now let A be the set of all |I| \times |I| matrices X=[x_{ij}], \ \ i,j \in I, \ x_{ij} \in R, such that x_{ij}=0 unless i \leq j. so the rows and columns of elements of A are indexed by I and the entries of

    them are in R. the addition and multiplication of elements of A is the usual addition and multiplication of matrixes. the identity matrix is clearly the identity element of A.

    note that if X=[x_{ij}], \ Y=[y_{ij}] are in A, and Z=XY=[z_{ij}], then for i \leq j we have z_{ij}=\sum_{i \leq k \leq j}x_{ik}y_{kj}, which is a finite sum since, as we assumed, the set [i,j] is finite. you still need

    to do a little work to see that A is closed under addition and multiplication. so A is an R algebra and we call A an incidence algebra. it's a nice and simple fact that an element X=[x_{ij}]

    of A is invertible if and only if x_{ii} is a unit in R for all i \in I. also note that if we label elements of I in a way that i comes before j whenever i \leq j, then every element of A would be an

    upper triangular matrix. so if I is totally ordered, then A would be exactly the ring of all |I| \times |I| upper triangular matrices with entries in R.


    i haven't seen incidence algebras in ring theory yet but it seems that they have important applications in discrete math.


    Edit:

    there's a book called "Incidence Algebras" and the authors are: Eugene Spiegel and Christopher O'Donnel. i can see from "google search book" that in this book incident algebras are studied

    in an abstract ring theory way, and so, i don't know about you, but i'm gonna get the book soon!
    Last edited by NonCommAlg; January 26th 2009 at 04:49 PM.
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