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.
Words in an alphabet with formal inverses, elementary cancellation, and reduced words
Definition
For a set , form a disjoint copy of formal inverses. A word on is a finite string of letters from ; the string of length zero is the empty word.
An elementary cancellation deletes two adjacent letters or . A word is reduced if no elementary cancellation applies. Words are freely equivalent if one can be transformed into the other by finitely many elementary cancellations and their reverse insertions. The reduction and uniqueness facts needed for the free-group construction are proved in Reduced words form the free group on an alphabet ↗.
Depends on
Used by
- Cyclically reduced words Definition
- Free group on a set of generators Definition
- The word-quotient model F_word(X):=W(X)/∼ with multiplication induced by concatenation Definition
- Reducing the word xx⁻¹yy⁻¹x leaves the reduced word x Example
- Every nonempty reduced word has the form tct⁻¹ with c nonempty and cyclically reduced Lemma
- Formal letters act by mutually inverse permutations on the set of reduced words Lemma
- Free equivalence is an equivalence relation and concatenation respects it Proposition
- Every class in W(X)/∼ contains exactly one reduced word Theorem
- Reduced words form the free group on an alphabet Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 13 results over 6 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
- McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)