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6.6 Mobius inversions

Let $X$ be a finite set.

Let ${\cal F} = \{f: X \times X \rightarrow {\cal R} ~|~$ if $f(x, y) \not= 0$, then 
$x \leq y \}$


Define the operation * on ${\cal F}$ by 
$$f*g = \cases{\Sigma_{\{z~|~ x \leq z \leq y \}} f(x, z)g(z, y) & if $x \leq y$ \cr 0 & otherwise}$$

Note $*$ is associative:  $f*(g*h) = (f*g)*h$

Let $\delta = \cases{1 & x = y \cr 0 & otherwise}$



Note $\delta$ acts as the identity for *:  $f* \delta = \delta * f = f$



If $f(x, x) \not= 0$ for all $x \in X$, then $f$ is invertible:  There exist $f^{-1}$ such that $f*f^{-1} = f^{-1}*f = \delta$.  In this case,


 $f^{-1}(x, x) = {1 \over f(x, x)}$

$f^{-1}(x, y) = -\Sigma_{\{z~:~ x \leq z < y\}}f^{-1}(x, z){f(z, y) \over f(y, y)}$ for $x < y$,

$f^{-1}(x, y) = 0$ if $x \not\leq y$.

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