False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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False: $μ_{P} (x, y)$ depends only on the cardinality of $[x, y]$

Statement

If two finite intervals have the same cardinality, then their endpoint Möbius values are equal.

Facts & Assumptions

Given: A four-element chain $c_{0} < c_{1} < c_{2} < c_{3}$ and the four-element diamond ${⊥, a, b, ⊤}$ with $⊥ < a < ⊤$ , $⊥ < b < ⊤$ , and $a, b$ incomparable.

[L1]

On a finite chain the endpoint value is $0$ whenever the interval contains an intermediate element (On a finite chain, the Möbius function is $1$ on the diagonal, $- 1$ on covers and $0$ on longer intervals).

[L2]

The diamond is the Boolean lattice on a two-element set, whose endpoint value is $(- 1)^{2} = 1$ (For $A \subseteq B$ in a finite Boolean lattice, $μ (A, B) = (- 1)^{∣ B ∖ A ∣}$ ).

Refutation

technique · direct

1.1

The endpoint interval $[c_{0}, c_{3}]$ has four elements and Möbius value $0$ by [L1].

1.2

The endpoint interval $[⊥, ⊤]$ in the diamond also has four elements but has Möbius value $1$ by [L2].

2.1

Equal interval cardinality therefore does not determine the Möbius value; the internal order structure matters.

step 1.1step 1.2∎

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)

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