ExampleConstruction: AI-adaptedVerification: Not suppliedaudited 2026-07-31

How statement and proof provenance work

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

Example

Let the diamond have bottom $⊥$ , top $⊤$ , and incomparable middle elements $a, b$ . It is the Boolean lattice on a two-element set, so For $A \subseteq B$ in a finite Boolean lattice, $μ (A, B) = (- 1)^{∣ B ∖ A ∣}$ gives

$μ (⊥, ⊥) = 1, μ (⊥, a) = μ (⊥, b) = - 1, μ (⊥, ⊤) = 1.$

The endpoint recurrence displays the same computation directly:

$1 - 1 - 1 + μ (⊥, ⊤) = 0,$

and hence $μ (⊥, ⊤) = 1$ (The Möbius recurrence: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ vanish when $x < y$ ).

Depends on

Used by

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

Dependency tree · two levels

14 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

R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)

The endpoint Möbius value of the four-element diamond is 11