MoebiusFunction

The Möbius function of a finite poset — the inverse of the zeta function in the incidence algebra. Specialising the poset recovers number theory's and inclusion–exclusion.

WikipediaIncidence algebra↗MathWorldMoebiusFunction↗WikidataQ371133↗

MoebiusFunction(poset, x, y) on the interval , and 0 when

Domain: Incidence algebras

Details

• Defined by and — which is exactly the statement that inverts
• On `DivisorLattice(n)`, is the classical number-theoretic
• On `BooleanLattice(n)`, — the signs of inclusion–exclusion
• On a chain, is 1 on a point, on a cover, and 0 on anything longer
• Posets: `Chain(n)`, `BooleanLattice(n)` (elements are subsets), `DivisorLattice(n)` (elements are divisors)

Examples

See also: MoebiusInvert, PosetZeta, PosetElements