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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

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 $δ$ .

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]

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

1.4

Likewise $(f * δ) (x, y) = \sum_{x \leq z \leq y} f (x, z) δ (z, y) = f (x, y)$ 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 $δ$ ; 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 · two levels

16 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)

Pointwise addition and convolution make I(P,R)I(P,R) a ring with identity δδ