DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

A directed set and an inverse system of groups indexed by it

Definition

A directed set is a nonempty preorder $(I, \leq)$ such that for every $i, j \in I$ there is a $k \in I$ with $i \leq k$ and $j \leq k$ (Directed preorders and nets).

An inverse system of groups indexed by $I$ is a family of groups $(G_{i})_{i \in I}$ together with homomorphisms

φ_{i j} : G_{j} \to G_{i} (i \leq j)

satisfying $φ_{i i} = {id}_{G_{i}}$ and $φ_{i k} = φ_{i j} \circ φ_{j k}$ whenever $i \leq j \leq k$ (Monoid homomorphism and group homomorphism).

Depends on

Used by

Dependency tree · two levels

8 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

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)