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.
Cyclically reduced words
Definition
A reduced word is cyclically reduced when it is empty or its first letter is not the formal inverse of its last letter. Equivalently, every cyclic rotation of the word is reduced.
If as a literal concatenation of words, the word is a cyclic permutation of . This includes itself by taking or empty.
Depends on
Used by
- A piece is a common initial segment occurring in two distinct places of a symmetrised relator set Definition
- Dehn-reduced words and Dehn presentations Definition
- Sc toolkit symmetrised relators and pieces Definition
- The symmetrisation of a relator set closes under inverses and cyclic conjugates Definition
- Every nonempty reduced word has the form tct⁻¹ with c nonempty and cyclically reduced Lemma
- Free groups are residually finite Theorem
- Free groups are torsion-free Theorem
- Two cyclically reduced words in a free group are conjugate if and only if one is a cyclic permutation of the other Theorem
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
- Alexei Myasnikov and Vladimir Shpilrain, Combinatorics over Free Groups, §2.2.1 (standard reference, not scraped)
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial Group Theory (standard reference, not scraped)