CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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 $P : = {⊥, ⊤} \cup (N \times {m})$ , with all three pieces disjoint, and the relation in which $⊥ < ⊤$ and $⊥ < (n, m) < ⊤$ for every $n \in N$ , distinct middle elements are incomparable, and equality is allowed.

[F1]

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

[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 $⊥ < (n, m) < ⊤$ , whose endpoints are already comparable; thus it is transitive. Hence $P$ is a poset.

given

1.2

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

givenF1F2

2.1

For constant-one functions $f$ and $g$ , the formal endpoint value is $(f * g) (⊥, ⊤) = \sum_{z \in P} 1_{R}$ . 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 $(⊥, ⊤)$ , refuting the statement.

step 1.2step 2.1∎

Remarks


\draw[gray!65] (bot)--(m0) (bot)--(m1) (bot)--(m2) (bot)--(mn) (m0)--(top) (m1)--(top) (m2)--(top) (mn)--(top); \draw[gray!55,densely dotted] (bot)--(dots)--(top);

\node[anchor=west] at (5.15,2.05) {countably many incomparable}; \node[anchor=west] at (5.15,1.55) {middle elements}; \node[anchor=west] at (5.15,.75) {$[\bot,\top]=P$ is infinite}; \end{tikzpicture} ```

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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)