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In Combinatorics the Rota Way Kung gives a proof of Phillip Hall's Theorem of chains;

If $P$ a poset and $(x,y)$ an interval on P then $\mu(x,y)=-c_1+c_2-c_3...$

Here $\mu$ is the Möbius function, $\zeta=\mu^{-1}$, $c_i$ counts the number of lenghth-i chains with x and y as endpoints (so $x < p_1 <...p_{i-1}<y$). The proof given is;

''Let $\eta=\zeta-\delta$ then we have $\mu=(\eta+\delta)^{-1}=*=\delta-\eta+\eta^2-\eta^3...$'' and we are done

So I buy everything except *. It looks like $\frac{1}{1-r}=\sum_0^{\infty}r^n$ but I quite connect them precisely.

EDIT;

Similar to; Geometric series of an operator

But how could I guarantee that $\eta$ is nilpotent? There are no restrictions on $P$..it could be (viscously) infinite (like $P=\mathbb{R}$). Surely the series dosn't converge for real intervals?

Muselive
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  • $\zeta$ is the inverse of the Möbius function and $\delta$ is the identity in the incidence algebra on $P$. I don't see why that should warrant a down vote, these are very standard symbols. https://en.wikipedia.org/wiki/Incidence_algebra – Muselive May 09 '21 at 11:38
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    Usually, when you talk about the incidence algebra of a poset $P$, it is assumed that $P$ is locally finite. E.g, the definition of incidence algebra on Wikipedia starts with this assumption.This condition would ensure that $\delta-\eta+\eta^2-\dots$ is well-defined. Are you sure you have not missed that condition? – Mike Earnest May 09 '21 at 17:22
  • Nope, It just says ''a poset P''. When he introduces incidence algebras however, P is indeed locally finite. I will chalk it up to poor editing (or perhaps reading comprehension on my part) – Muselive May 09 '21 at 17:32
  • I just looked at the book, I agree that it is unclear. It must be the case that $P$ is locally finite, else $-c_1+c_2-c_3+\dots$ would not even converge. – Mike Earnest May 09 '21 at 17:41

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