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The integer-valued Möbius function of a locally finite poset
Definition
Let be a locally finite poset. Take coefficients in the commutative ring (The integers form a commutative ring). The zeta incidence function has diagonal value , 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 is its unique inverse
so
with and as in The delta and zeta incidence functions. Its value is therefore an integer for every .
Remarks
The coefficient ring is fixed as . When a formula takes values in another ring , the integer acts through its canonical repeated-addition multiple of ; no characteristic-dependent second Möbius function is introduced.
Depends on
Used by
- A parabolic quotient interval of S4 whose Möbius value is 0, so the Eulerian sign formula does not extend to quotients Counterexample
- Deleted-position labels from a fixed reduced expression, the lexicographic shelling criterion, and Möbius data Definition
- A rank-three chain labeling translated into facets of the order complex Example
- The Möbius value of the rank-three interval [e,c] in S4 from the recurrence, with the parity and falling-chain checks Example
- Lexicographic chain shelling and the falling-chain Möbius formula Lemma
- The Möbius recurrence: μ_P(x,x)=1 and both interval sums of μ_P vanish when x<y Lemma
- Bruhat intervals are Eulerian: parity balance of the elements, and the Möbius function of a full interval Theorem
- The number-theoretic Möbius function is the poset Möbius function of divisibility: μ(n)=μ_∣(1,n) Theorem
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)