Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
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.

Associated graded algebra of a filtered algebra

Definition

Let A be a unital algebra with an increasing multiplicative filtration

0=F−1A⊆F0A⊆F1A⊆⋯ ,

where 1∈F0A and (FmA)(FnA)⊆Fm+nA. Its associated graded algebra is

gr⁡FA=⨁n≥0FnA/Fn−1A,

with multiplication on homogeneous classes

(a+Fm−1A)(b+Fn−1A)=ab+Fm+n−1A.

This product is well-defined. Indeed, if a changes by u∈Fm−1A and b by v∈Fn−1A, the product changes by ub+av+uv∈Fm+n−1A. Associativity and the unit class descend from A.

Depends on

Used by

Dependency tree · two levels

5 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