DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

Cyclically Britton-reduced HNN words

Definition

Let

$w = a_{0} t^{ε_{1}} a_{1} \dots t^{ε_{n}} a_{n}$

be a Britton-reduced HNN word in the sense of HNN words, pins, and Britton-reduced words. It is cyclically Britton-reduced when either $n = 0$ , or $n > 0$ and the cyclic rotation

$t^{ε_{n}} (a_{n} a_{0}) t^{ε_{1}} a_{1} \dots t^{ε_{n - 1}} a_{n - 1}$

is still Britton-reduced.

Equivalently, a Britton-reduced word of positive stable-letter length is cyclically Britton-reduced exactly when no pin appears across the two ends: if $ε_{n} = 1$ and $ε_{1} = - 1$ , then $a_{n} a_{0} \notin C_{-}$ ; if $ε_{n} = - 1$ and $ε_{1} = 1$ , then $a_{n} a_{0} \notin C_{+}$ .

Depends on

HNN words, pins, and Britton-reduced words

Used by

Dependency tree · two levels

2 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

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory (standard reference, not scraped)