Editing Möbius inversion formula (section)

==On posets==
{{See also|Incidence algebra}}
For a [[Partially ordered set|poset]] {{mvar|P}}, a set endowed with a partial order relation <math>\leq</math>, define the Möbius function <math>\mu</math> of {{mvar|P}} recursively by 

:<math>\mu(s,s) = 1 \text{ for } s \in P, \qquad \mu(s,u) = - \sum_{s \leq t < u} \mu(s,t), \quad \text{ for } s < u \text{ in } P.</math>

(Here one assumes the summations are finite.) Then for <math>f,g: P \to K</math>, where {{mvar|K}} is a commutative ring, we have 

:<math>g(t) = \sum_{s \leq t} f(s)  \qquad \text{ for all } t \in P</math>

if and only if 

:<math>f(t) = \sum_{s \leq t} g(s)\mu(s,t) \qquad \text{ for all }t \in P.</math>

(See Stanley's ''Enumerative Combinatorics'', Vol 1, Section 3.7.) The classical arithmetic Mobius function is the special case of the poset ''P'' of positive integers ordered by [[Divisor|divisibility]]: that is, for positive integers ''s, t,'' we define the partial order <math>s \preccurlyeq t  </math> to mean that ''s'' is a divisor of ''t''.