TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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An incidence function is convolution-invertible if and only if every diagonal value is a unit

Statement

Let $P$ be any locally finite poset, possibly infinite, let $R$ be a commutative ring, and let $f \in I (P, R)$ . Then $f$ is invertible under convolution if and only if $f (x, x)$ is a unit of $R$ for every $x \in P$ .

Facts & Assumptions

Given: A locally finite poset $P$ , a commutative ring $R$ , and $f \in I (P, R)$ .

[L1]

If every diagonal value of $f$ is a unit, the recursive interval formulas construct a two-sided convolution inverse (If every diagonal value of an incidence function is a unit, recursive interval formulas construct both a left and a right convolution inverse).

[F1]

An invertible element has a two-sided inverse (Left inverse, right inverse, and invertible element of a monoid), and a unit in a ring has a unique inverse (The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times}$ is a group under multiplication; $0 \in R^{\times}$ only in the zero ring).

[F2]

The convolution identity satisfies $δ (x, x) = 1_{R}$ (The delta and zeta incidence functions).

Proof

technique · direct

1.1

Suppose $f$ has a convolution inverse $g$ . Evaluating $f * g = δ$ at $(x, x)$ gives $f (x, x) g (x, x) = 1_{R}$ , and evaluating $g * f = δ$ gives $g (x, x) f (x, x) = 1_{R}$ . Thus $f (x, x)$ is a unit for every $x$ .

F1F2

1.2

Conversely, if every $f (x, x)$ is a unit, [L1] constructs a two-sided convolution inverse of $f$ .

2.1

Steps 1.1 and 1.2 prove both directions of the criterion.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

19 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)
Y. Guan and Y. Zhang, Additive Biderivations of Incidence Algebras, §2.1 (standard reference, not scraped)
Hameister–Rao–Simpson, Proposition 2.8 (standard reference, not scraped)