CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

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Both forms of Möbius inversion hold on every finite poset

Statement

If the ground set of a poset $P$ is finite, then $P$ is both lower-finite and upper-finite. Consequently both forms of Möbius inversion on a lower-finite poset, with the dual upper-finite form hold for functions from $P$ into any commutative ring.

Facts & Assumptions

Given: A poset $P$ with finite ground set.

[L1]

Every subset of a finite set is finite (A subset of a finite set is finite, with $∣ B ∣ \leq ∣ A ∣$ , and equality holds if and only if $B = A$ ).

[L2]

Lower-finite and upper-finite Möbius inversion hold under their respective hypotheses (Möbius inversion on a lower-finite poset, with the dual upper-finite form).

Proof

technique · direct

1.1

For each $y \in P$ , the principal ideal $P_{\leq y}$ is a subset of $P$ , hence finite by [L1]; thus $P$ is lower-finite.

1.2

For each $x \in P$ , the principal filter $P_{\geq x}$ is a subset of $P$ , hence finite by [L1]; thus $P$ is upper-finite.

2.1

Applying the two separate parts of [L2] proves both inversion formulas on $P$ .

step 1.1step 1.2L2∎

Depends on

Used by

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

Dependency tree · two levels

20 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

F. Gotti, Incidence Algebras, MIT 18.211 notes (standard reference, not scraped)
R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)