what is incidence algebra? what does it include? what is its applications? anyone send me lecture notes?
let $\displaystyle R$ be a commutative domain and $\displaystyle (I, \leq)$ a (not necessarily finite) partially ordered set with this property that for any $\displaystyle i,j \in I$ with $\displaystyle i \leq j,$ the set: $\displaystyle [i,j]=\{k \in I: \ i \leq k \leq j \}$ is finite.
now let $\displaystyle A$ be the set of all $\displaystyle |I| \times |I|$ matrices $\displaystyle X=[x_{ij}], \ \ i,j \in I, \ x_{ij} \in R,$ such that $\displaystyle x_{ij}=0$ unless $\displaystyle i \leq j.$ so the rows and columns of elements of $\displaystyle A$ are indexed by $\displaystyle I$ and the entries of
them are in $\displaystyle R.$ the addition and multiplication of elements of $\displaystyle A$ is the usual addition and multiplication of matrixes. the identity matrix is clearly the identity element of $\displaystyle A.$
note that if $\displaystyle X=[x_{ij}], \ Y=[y_{ij}]$ are in $\displaystyle A,$ and $\displaystyle Z=XY=[z_{ij}],$ then for $\displaystyle i \leq j$ we have $\displaystyle z_{ij}=\sum_{i \leq k \leq j}x_{ik}y_{kj},$ which is a finite sum since, as we assumed, the set $\displaystyle [i,j]$ is finite. you still need
to do a little work to see that $\displaystyle A$ is closed under addition and multiplication. so $\displaystyle A$ is an $\displaystyle R$ algebra and we call $\displaystyle A$ an incidence algebra. it's a nice and simple fact that an element $\displaystyle X=[x_{ij}]$
of $\displaystyle A$ is invertible if and only if $\displaystyle x_{ii}$ is a unit in $\displaystyle R$ for all $\displaystyle i \in I.$ also note that if we label elements of $\displaystyle I$ in a way that $\displaystyle i$ comes before $\displaystyle j$ whenever $\displaystyle i \leq j,$ then every element of $\displaystyle A$ would be an
upper triangular matrix. so if $\displaystyle I$ is totally ordered, then $\displaystyle A$ would be exactly the ring of all $\displaystyle |I| \times |I|$ upper triangular matrices with entries in $\displaystyle R.$
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!