I stumbled across a paper by Gian-Carlo Rota last night that deals with incidence algebras. I haven’t finished reading it yet, but it looks like it has some insight into what’s going on here. If anyone wants to take a look at it, here’s the link.

dedekind.mit.edu/~rstan/pubs/pubfiles/10.pdf

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*Related*

The link is broken.

Related to the topic of incidence algebras and posets is Mobius functions –as was pointed out in class at some point.

A nice resource on this and many, many more topics having to do with posets and lattices, is the textbook .

In particular, here is something interesting having to do with the incidence algebra of a poset. For those of you who like linear algebra and matrices, you will be excited to learn that the convolution, the multiplication we defined for the incidence algebra, in the same thing as multiplication of upper triangular matrices. Further, the elements of the incidence algebra can also be realized as matrices.