# “Double convolution” with the Mobius function on a poset

Let $f$ and $g$ be arbitrary (say integer-valued) functions on some poset $P$, and say $\mu$ is the Mobius function of $P$. I'm studying a quantity that's a sort of "double convolution" of $f$ and $g$ with $\mu$: $$e(f,g):=\sum_{x\in P}\sum_{y\in P}f(x)\mu(x,y)g(y).$$ Specifically, I'm interested in the behavior of $e(f,g)$ when I restrict to some subset of $P$. An example of the sort of result I'm after is the following thing that I already proved:

For $z\in P$, let $e_z(f,g)$ be the number you get by taking the definition of $e$ but summing over $P-\{z\}$ and using the Mobius function for $P-\{z\}$. Then $$e(f,g)-e_z(f,g)=\left(\sum_{x\le z}f(x)\mu(x,z)\right)\left(\sum_{y\ge z}\mu(z,y)g(y)\right).$$

I'm not very well-versed in the combinatorics of posets, but the proof of this fact was straightforward and kind of pretty, which led me to suspect that there might already be a nice theory of the sorts of quantities I'm looking at. In addition to being very helpful for the project I'm working on, it would just be fun to know whether someone's studied these guys before. It would be great, for example, if there were some nice way to express what happens in the above case when you remove a whole bunch of sets at once. Does anyone know if this is well-studied anywhere?

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