Alphabeta MathHow 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.
The delta and zeta incidence functions
Definition
Let be locally finite and let be a commutative ring with zero and identity (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides). The delta function and zeta function in (The incidence functions of a locally finite poset and their convolution) are
Both are functions on the comparable pairs of . The delta function is supported on the diagonal, while the zeta function is constant on every interval.
Depends on
Used by
- On a two-element chain, an incidence function with a zero diagonal value is not convolution-invertible Counterexample
- The integer-valued Möbius function μ_P of a locally finite poset Definition
- The Möbius recurrence: μ_P(x,x)=1 and both interval sums of μ_P vanish when x<y Lemma
- An incidence function is convolution-invertible if and only if every diagonal value is a unit Theorem
- Pointwise addition and convolution make I(P,R) a ring with identity δ Theorem
Dependency tree · two levels
13 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)