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Incidence Algebras and Möbius Inversion

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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Intervals in a poset; locally finite, lower-finite and upper-finite posets

Definition

The incidence functions I(P,R) of a locally finite poset and their convolution

Definition

The delta and zeta incidence functions

Lemma

Incidence convolution is associative and distributes over pointwise addition

Theorem

Pointwise addition and convolution make I(P,R) a ring with identity δ

Lemma

If every diagonal value of an incidence function is a unit, recursive interval formulas construct both a left and a right convolution inverse

Theorem

An incidence function is convolution-invertible if and only if every diagonal value is a unit

Definition

The integer-valued Möbius function μ_P of a locally finite poset

Lemma

The Möbius recurrence: μ_P(x,x)=1 and both interval sums of μ_P vanish when x<y

Theorem

Möbius inversion on a lower-finite poset, with the dual upper-finite form

Corollary

Both forms of Möbius inversion hold on every finite poset

Theorem

The Möbius function of a product poset is the product of the Möbius functions

Theorem

For A⊆ B in a finite Boolean lattice, μ(A,B)=(-1)^| B∖ A|

Corollary

The complementary inclusion-exclusion formula is Möbius inversion on the Boolean lattice

Theorem

On a finite chain, the Möbius function is 1 on the diagonal, -1 on covers and 0 on longer intervals

Definition

The divisibility poset of positive integers

Lemma

The divisibility poset is lower-finite, and each divisor interval factorises as a product of finite chains of prime exponents

Definition

The number-theoretic Möbius function μ(n) from prime factorisation

Theorem

The number-theoretic Möbius function is the poset Möbius function of divisibility: μ(n)=μ_∣(1,n)

Corollary

Classical Möbius inversion over positive divisors

Corollary

The number-theoretic Möbius function is multiplicative on coprime positive integers

False statement

False: convolution defines an incidence algebra for every poset

False statement

False: μ_P(x,y) depends only on the cardinality of [x,y]

False statement

False: classical Möbius inversion and inclusion-exclusion are unrelated inversion principles

Sources