DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-07-31

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The integer-valued Möbius function $μ_{P}$ of a locally finite poset

Definition

Let $P$ be a locally finite poset. Take coefficients in the commutative ring $Z$ (The integers form a commutative ring). The zeta incidence function has diagonal value $1$ , hence is convolution-invertible by An incidence function is convolution-invertible if and only if every diagonal value is a unit. The Möbius function of $P$ is its unique inverse

$μ_{P} : = ζ^{- 1} \in I (P, Z),$

$μ_{P} * ζ = δ = ζ * μ_{P}$

with $δ$ and $ζ$ as in The delta and zeta incidence functions. Its value $μ_{P} (x, y)$ is therefore an integer for every $x \leq y$ .

Remarks

The coefficient ring is fixed as $Z$ . When a formula takes values in another ring $R$ , the integer $μ_{P} (x, y)$ acts through its canonical repeated-addition multiple of $1_{R}$ ; no characteristic-dependent second Möbius function is introduced.

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

F. Gotti, Incidence Algebras, MIT 18.211 notes (standard reference, not scraped)
R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)
Y. Guan and Y. Zhang, Additive Biderivations of Incidence Algebras, §2.1 (standard reference, not scraped)

The integer-valued Möbius function μPμP​ of a locally finite poset