TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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Pointwise addition and convolution make $I(P,R)$ a ring with identity $\delta$

Statement

If $P$ is locally finite and $R$ is a commutative ring, then pointwise addition and incidence convolution make $I(P,R)$ a ring whose multiplicative identity is the delta incidence function $\delta$ .

Facts & Assumptions

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

[L1]

All functions from a set into a ring form an abelian group under pointwise addition, with pointwise zero and additive inverses (The ring $R^{X}$ of all functions from a set $X$ into a ring, with pointwise operations, Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

[L2]

Incidence convolution is associative and distributes over pointwise addition on both sides (Incidence convolution is associative and distributes over pointwise addition).

[F1]

$\delta(x,y)$ is $1_R$ on the diagonal and $0_R$ off it (The delta and zeta incidence functions).

Proof

technique · direct

1.1

Since $I(P,R)$ is the set of functions from the comparable pairs of $P$ to $R$ , [L1] makes it an abelian group under pointwise addition.

1.2

Associativity of convolution and both distributive laws are [L2].

1.3

For $x\le y$ , $(\delta*f)(x,y)=\sum_{x\le z\le y}\delta(x,z)f(z,y)=f(x,y)$ because only the term $z=x$ is nonzero.

1.4

Likewise $(f*\delta)(x,y)=\sum_{x\le z\le y}f(x,z)\delta(z,y)=f(x,y)$ because only the term $z=y$ is nonzero.

2.1

Thus convolution is associative, distributes over the pointwise abelian-group operation, and has the two-sided identity $\delta$ ; these are exactly the ring axioms.

step 1.1step 1.2step 1.3step 1.4L1∎

Depends on

Used by

If every diagonal value of an incidence function is a unit, recursive interval formulas construct both a left and a right convolution inverse Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 16 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)

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