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CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-07-31
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A four-element chain and a four-element diamond have equal-size endpoint intervals but Möbius values 0 and 1

Statement refuted

The endpoint Möbius value of a finite interval is determined by the number of elements in that interval (False: μP(x,y) depends only on the cardinality of [x,y]).

Facts & Assumptions

Given: The four-element chain c0<c1<c2<c3 and the four-element diamond with endpoints ⊥<⊤.

[F1]

The chain computation gives μ(c0,c3)=0 (The Möbius table of a four-element chain).

[F2]

The diamond computation gives μ(⊥,⊤)=1 (The endpoint Möbius value of the four-element diamond is 1).

Counterexample

technique · direct
1.1

Both endpoint intervals have four elements.

given
1.2

Their endpoint Möbius values are nevertheless 0 and 1 by [F1] and [F2].

F1F2
2.1

Hence equal-size intervals can have different Möbius values, and the statement is false.

step 1.1step 1.2∎

Remarks


\node[elt] (c0) at (-2.7,0) {$c_0$}; \node[elt] (c1) at (-2.7,1.05) {$c_1$}; \node[elt] (c2) at (-2.7,2.1) {$c_2$}; \node[elt] (c3) at (-2.7,3.15) {$c_3$}; \draw[gray!70] (c0)--(c1)--(c2)--(c3);

\node[elt] (bot) at (2.7,0) {$\bot$}; \node[elt] (a) at (1.65,1.55) {$a$}; \node[elt] (b) at (3.75,1.55) {$b$}; \node[elt] (top) at (2.7,3.15) {$\top$}; \draw[gray!70] (bot)--(a)--(top) (bot)--(b)--(top);

\node[font=\small] at (-2.7,-.75) {$\mu(c_0,c_3)=0$}; \node[font=\small] at (2.7,-.75) {$\mu(\bot,\top)=1$}; \end{tikzpicture} ```

Depends on

Used by

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Dependency tree · two levels

6 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