1. ## anyone help me about incidence algebra?

what is incidence algebra? what does it include? what is its applications? anyone send me lecture notes?

2. Originally Posted by bogazichili

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!

3. here.