[math] \newcommand{\mathds}{\mathbb}[/math]

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Establish the Möbius inversion formula, namely

[[math]] f(\sigma)=\sum_{\pi\leq\sigma}g(\pi) \quad\implies\quad g(\sigma)=\sum_{\pi\leq\sigma}\mu(\pi,\sigma)f(\pi) [[/math]]

for the functions on [math]P(p)[/math].