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

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

Statement

Let $P$ be locally finite, let $R$ be a commutative ring, and let $f\in I(P,R)$ . Suppose $f(x,x)$ is a unit of $R$ for every $x\in P$ . Then the recursive formulas

$g(x,x)=f(x,x)^{-1},\qquad g(x,y)=-f(x,x)^{-1}\sum_{x<z\le y}f(x,z)g(z,y)\quad(x<y)$

and

$h(x,x)=f(x,x)^{-1},\qquad h(x,y)=-\left(\sum_{x\le z<y}h(x,z)f(z,y)\right)f(y,y)^{-1}\quad(x<y)$

define incidence functions satisfying $f*g=\delta$ and $h*f=\delta$ . They coincide, so their common value is a two-sided convolution inverse of $f$ .

Facts & Assumptions

Given: A locally finite poset $P$ , a commutative ring $R$ , and an incidence function $f$ whose diagonal values are units.

[L1]

Strong induction: if a property at $n$ follows from its truth at every smaller natural, it holds for every natural (Strong (complete) induction).

[F1]

Every interval $[x,y]$ is finite; if $x<z\le y$ , then $[z,y]$ is a proper subset of $[x,y]$ , and if $x\le z<y$ , then $[x,z]$ is a proper subset (Intervals in a poset; locally finite, lower-finite and upper-finite posets).

[L2]

A proper subset of a finite set has strictly smaller finite cardinality (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ , The cardinality $\lvert A\rvert$ of a finite set).

[L3]

A unit of a ring has a unique inverse, and the units form a group (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).

[L4]

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

[F2]

Finite sums over the displayed subintervals are defined in the additive commutative monoid of $R$ (A finite sum in a commutative monoid indexed by an arbitrary finite set).

Proof

technique · induction

1.1

On a diagonal interval the equations $(f*g)(x,x)=1_R$ and $(h*f)(x,x)=1_R$ force $g(x,x)=h(x,x)=f(x,x)^{-1}$ by [L3].

baseL3

1.2

Fix a natural $n$ and assume that $g$ and $h$ have been uniquely defined on every interval of cardinality less than $n$ , with the required convolution equations there.

2.1

Let $x<y$ with $|[x,y]|=n$ . Every $g(z,y)$ occurring in $\sum_{x<z\le y}f(x,z)g(z,y)$ belongs to the proper subinterval $[z,y]$ , and every $h(x,z)$ in $\sum_{x\le z<y}h(x,z)f(z,y)$ belongs to the proper subinterval $[x,z]$ ; their cardinalities are less than $n$ by [F1] and [L2].

step 1.2F1L2

3.1

The displayed formulas in the Statement therefore assign unique values to $g(x,y)$ and $h(x,y)$ , since the sums are finite and both diagonal inverses are unique.

step 2.1F2L3construct

4.1

Isolating the term $z=x$ in convolution gives $(f*g)(x,y)=f(x,x)g(x,y)+\sum_{x<z\le y}f(x,z)g(z,y)=0_R$ by the defining formula for $g(x,y)$ .

step 3.1L3

4.2

Isolating the term $z=y$ gives $(h*f)(x,y)=\sum_{x\le z<y}h(x,z)f(z,y)+h(x,y)f(y,y)=0_R$ by the defining formula for $h(x,y)$ .

step 3.1L3

5.1

Steps 1.1 through 4.2, with strong induction on $|[x,y]|$ , define $g$ and $h$ on every comparable pair and give $f*g=\delta$ and $h*f=\delta$ .

step 1.1step 1.2step 2.1step 3.1step 4.1step 4.2L1discharge-induction

6.1

Associativity and the identity law in [L4] now give $h=h*\delta=h*(f*g)=(h*f)*g=\delta*g=g$ .

step 5.1L4

7.1

Hence the two recursive one-sided inverses coincide and their common value is a two-sided convolution inverse of $f$ .

step 5.1step 6.1discharge-induction∎

Depends on

Used by

An incidence function is convolution-invertible if and only if every diagonal value is a unit Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 21 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)
Hameister–Rao–Simpson, Proposition 2.8 (standard reference, not scraped)