CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-04 (gpt-5.6-sol-codex-subscription)

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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

Statement refuted

Convolution of incidence functions is defined on every poset, even without local finiteness (False: convolution defines an incidence algebra for every poset).

Facts & Assumptions

Given: A nonzero commutative ring $R$ , the set $P:=\{\bot,\top\}\cup(\mathbb N\times\{m\})$ , with all three pieces disjoint, and the relation in which $\bot<\top$ and $\bot<(n,m)<\top$ for every $n\in\mathbb N$ , distinct middle elements are incomparable, and equality is allowed.

[F1]

$\mathbb N$ is the infinite set of natural numbers (The natural numbers $\mathbb{N}$ (von Neumann), Finite, countably infinite, countable, uncountable, The pigeonhole principle on $\mathbb{N}$ ).

[F2]

A poset is locally finite exactly when every interval is finite (Intervals in a poset; locally finite, lower-finite and upper-finite posets).

[F3]

Incidence convolution is the finite ring sum $(f*g)(x,y)=\sum_{z\in[x,y]}f(x,z)g(z,y)$ ; local finiteness is what makes this sum defined (The incidence functions $I(P,R)$ of a locally finite poset and their convolution).

Counterexample

technique · direct

1.1

The relation is reflexive. Opposite inequalities force equality, so it is antisymmetric, and its only nontrivial two-step strict chains have the form $\bot<(n,m)<\top$ , whose endpoints are already comparable; thus it is transitive. Hence $P$ is a poset.

given

1.2

Every point of $P$ lies between $\bot$ and $\top$ , so $[\bot,\top]=P$ . It contains the countably infinite subset $\mathbb N\times\{m\}$ , hence is infinite and $P$ is not locally finite.

givenF1F2

2.1

For constant-one functions $f$ and $g$ , the formal endpoint value is $(f*g)(\bot,\top)=\sum_{z\in P}1_R$ . This has one term for every natural-indexed middle point, in addition to the endpoints, and is not the finite ring sum required by convolution.

step 1.2F3

3.1

Thus the proposed convolution is not defined at $(\bot,\top)$ , refuting the statement.

step 1.2step 2.1∎

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 58 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

F. Gotti, Incidence Algebras, MIT 18.211 notes (standard reference, not scraped)
Y. Guan and Y. Zhang, Additive Biderivations of Incidence Algebras, §2.1 (standard reference, not scraped)