Is it possible or interesting to try to build Mobius inversion for non-locally finite posets? In some of my free time I've been reading up a bit on Mobius inversion. What puzzles me is that in just about every discussion of incidence algebras that I can find there is the strict requirement of a locally finite poset $\mathbb{P}$, meaning that for every pair of elements $x,y \in \mathbb{P}$ the interval $[x,y]$ between them is finite. (Recall $[x,y]=\{z\in\mathbb{P}: x\leq z\leq y\}$.)
Now it's clear to me that this condition is to ensure that multiplication $\displaystyle f\ast g = \sum f(x,z)g(z,y)[x\leq z\leq y]$ is defined everywhere, where $[\phi(\vec{v})]$ is the Iverson bracket. But as any analysis student knows, infinite sums can certainly converge in an appropriate topology. So why not try to extend this for non-locally finite posets? Is there some weird technical problem with trying to construct infinite sums on posets that I'm not thinking of?

Mobius inversion just doesn't make sense for non-locally finite posets. Let's take the simplest nontrivial example: a poset with a bottom element $\bot$, a top element $\top$, and countably many elements sandwiched between them $\mathbb{N}$. Now check the recursive definition of the Mobius function: to evaluate $\mu(\bot, \top)$ you need to take an infinite sum of $1$s, which doesn't converge in any reasonable sense.
It's true that convolution can make sense in more generality than the locally finite case; for example, we can define convolution on an arbitrary poset... as long as we're willing to restrict our attention to functions with finite support. But this means the zeta function is no longer an element of the convolution algebra in general, so we no longer have an element to try to invert to get Mobius inversion.
