now let be the set of all matrices such that unless so the rows and columns of elements of are indexed by and the entries of
them are in the addition and multiplication of elements of is the usual addition and multiplication of matrixes. the identity matrix is clearly the identity element of
note that if are in and then for we have which is a finite sum since, as we assumed, the set is finite. you still need
to do a little work to see that is closed under addition and multiplication. so is an algebra and we call an incidence algebra. it's a nice and simple fact that an element
of is invertible if and only if is a unit in for all also note that if we label elements of in a way that comes before whenever then every element of would be an
upper triangular matrix. so if is totally ordered, then would be exactly the ring of all upper triangular matrices with entries in
i haven't seen incidence algebras in ring theory yet but it seems that they have important applications in discrete math.
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!