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Incidence Algebras and Möbius Inversion — Examples

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

3 · Items

Example

The full Möbius table of the Boolean lattice 2^[3]

Example

The Möbius table of a four-element chain

Example

The Möbius function on the divisor poset of 12 and its agreement with μ(1),μ(2),μ(3),μ(4),μ(6),μ(12)

Example

The endpoint Möbius value of the four-element diamond is 1

Example

Möbius inversion of ∑_d∣ nφ(d)=n gives φ(n)=∑_d∣ nμ(d)(n/d)

Counterexample

A poset with a bottom, a top and countably many incomparable middle elements has an infinite interval, so convolution of constant-one functions is not defined

Counterexample

On a two-element chain, an incidence function with a zero diagonal value is not convolution-invertible

Counterexample

A four-element chain and a four-element diamond have equal-size endpoint intervals but Möbius values 0 and 1

Sources