Super incomplete thought: how does this point of view relate to Euler characteristic? Can I get to Euler characteristic by asking how to solve an equation for some quantities q in terms of some quantities Q?
Author of the post here: There is quite a deep connection actually. You can assign a simplicial complex to a partial order P with a max and a min element (0 and 1). Then the Möbius function on P calculates the (reduced) Euler characteristic of that simplicial complex as µ(0, 1)=\Chi. For example, if the partial order is a power set mereology (a Boolean algebra) on 3 elements, then the associated simplicial complex is a triangle, and µ(0, 1) = µ(\emptyset, {a, b, c})=(-1)^3, which is the correct answer as a triangle (without interior) is homeomorphic to a circle.
In a way, calculating quantities q through Möbius inversion is just calculating Euler characteristics, weighted by by Q. (with some caveats)
In this article, we fix a mereology and a kind of quantity Q that "decomposes" over it---in the sense that Q(p) = sum_{r <= p} q(r) for some function q(r)---and then see that Mobius inversion lets us solve for q in terms of Q. In terms of incidence algebras, we're saying: assume Q = zeta q, as a product of elements in an incidence algebra. Then zeta has an inverse mu, so q = mu Q.
In other situations, we might want to "solve for" a quantity Q that decomposes over some class of metrologies while respecting some properties. The "simpler" and more "homogeneous" the parts of your mereology, the less you can express, but the easier it becomes to reason about Q. A mereology that breaks me up into the empty set, singleton sets with each of my atoms, and the set of all my atoms admits no "decomposing quantities" besides a histogram of my atoms. An attempt to measure "how healthy I am" in terms of that mereology can't do much. On the other hand, if I choose the mereology that breaks me up into the empty set and my whole, all quantities decompose but I have no tools to reason about them.
I guess Euler characteristic could be an example of how the requirement of respecting a certain kind of mereology can "bend" a hard-to-decompose quantity into a weirder but "nicer" quantity. For example, say we're interested in defining a Q that attempts to "count the number of connected regions" of some object, and we insist on using a mereology that lets us divide regions up into "cells". Of course this is impossible, as we can see in the problem of counting connected components of a graph-like object: we can't get the answer just as a function of the number of vertices and edges. However, if we insist on assigning a value of 1 to "blobs" of any dimension, the "compositionality requirement" forces us to define the Euler characteristic. This doesn't help us much with graph algorithms in general, but gives us an unexpectedly easy way to, say, count the number of blob-shaped islands on a map.
In a way, calculating quantities q through Möbius inversion is just calculating Euler characteristics, weighted by by Q. (with some caveats)