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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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 25 results over 14 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)